Equiaffine geometry of level sets and ruled hypersurfaces with equiaffine mean curvature zero
aa r X i v : . [ m a t h . DG ] N ov EQUIAFFINE GEOMETRY OF LEVEL SETS AND RULED HYPERSURFACESWITH EQUIAFFINE MEAN CURVATURE ZERO
DANIEL J. F. FOX
Abstract.
Basic aspects of the equiaffine geometry of level sets are developed systematically.As an application there are constructed families of 2 n -dimensional nondegenerate hypersurfacesruled by n -planes, having equiaffine mean curvature zero, and solving the affine normal flow.Each carries a symplectic structure with respect to which the ruling is Lagrangian. Introduction
The goal of this article is the construction of 2 n -dimensional equiaffine mean curvature hyper-surfaces ruled by totally geodesic n -planes. To achieve this aim, basic aspects of the equiaffinegeometry of level sets are developed systematically. In the process there are made some remarksabout the equiaffine geometry of level sets and the construction of the affine normal distribution inspaces more general than flat affine space.The examples constructed generalize to higher dimensions equiaffine mean curvature zero ruledsurfaces constructed by A. Mart´ınez and F. Milan in [12]. Precisely, Theorem 3.4 shows that,given Q ∈ C ∞ ( M ) and a centroaffine immersion A : M → R n +1 such that the pullback of thecontraction of the radial Euler field with the volume form on R n +1 is a nonzero constant multipleof the volume form du ∧ · · · ∧ du n for some flat affine coordinates on M , then, with respect to anappropriate equiaffine structure on M × R n +1 ∗ , the level sets of the function F : M × R n +1 ∗ → R defined by F ( u, x ) = h A ( u ) , x i + Q ( u ) are equiaffine mean curvature zero hypersurfaces foliatedby n -dimensional affine planes isotropic with respect to the equiaffine metric. Moreover each levelset carries a symplectic structure with respect to which the leaves of the ruling are Lagrangian.By Lemma 3.2, in general a level set of F is not an improper affine sphere, for this is the caseif and only if the image of A is contained in a hyperplane. Finally, there is an explicit smoothmap ϕ : R × R n → M × R n +1 such that F ( ϕ ( t, p )) is a constant multiple of t , so that ϕ ( t, · )parameterizes part of a level set of F , and that solves the affine normal flow in the sense that ddt ϕ ( t, p ) is equal to the affine normal W ϕ ( t,p ) at p to the image ϕ ( t, R n ).In the case n = 2, taking Q = 0 and taking as the components of A linearly independent solutionsof a homogeneous linear second order differential equation yields examples in [12]. The prototypicalexample is a level set Σ t = { ( u, x, y ) : F ( u, x, y ) = t } of F ( u, x, y ) = x sin u + y cos u . The equiaffinenormal is − sin u∂ x − cos u∂ y , the symplectic form is the restriction of du ∧ (cos udx − sin udy ), andthe ruling is generated by cos u∂ x − sin u∂ y . The level set Σ t is a helicoid with the parameterization { ( r, s sin r + t cos r, − s cos r + t sin r ) : ( r, s ) ∈ R } .In section 3.2, several recipes for explicitly constructing centroaffine immersions with the nec-essary properties are given. A typical example obtained from these constructions is that, for anysmooth function Q ( u, v ) of two variables, the level sets in R of the function F ( x , x , x , x , x )= x sech x cos( x cosh x ) + x sech x sin( x cosh x ) + x tanh x + Q ( x , x )(1.1)are smoothly immersed equiaffine mean curvature zero hypersurfaces ruled by 2-planes. In flat affine space any two parallel metrics of the same signature are affinely equivalent. Whilethe Gauss-Kronecker curvature of a hypersurface with respect to different parallel metrics changes,the condition that it vanish or not does not; a hypersurface is nondegenerate if its Gauss-Kroneckercurvature is everywhere nonzero with respect to any parallel metric (equivalently, the second fun-damental form is nondegenerate), and similarly, it makes sense to speak of a hypersurface withzero Gauss-Kronecker curvature with respect to any parallel metric (a developable hypersurface).A ruling of a hypersurface by k -planes is a foliation by totally geodesic k -dimensional submanifolds,and a hypersurface equipped with a ruling is said to be ruled . A ruling is cylindrical if its leavesare parallel planes. A cylindrical ruling is developable because the vectors tangent to the rulinglie in the radical of the second fundamental form. However, there exist developable hypersurfacesthat are not cylindrical. By construction the level sets of the the functions F of Section 3 are ruledbut not developable. On the other hand, the graph of F is a smoothly immersed noncylindricaldevelopable (Gauss-Kronecker curvature zero) hypersurface.Because the plane tangent to a ruling is isotropic with respect to the second fundamental form,the second fundamental form of a ruled hypersurface has indefinite signature. If the hypersurfaceis nondegenerate then the rank of a ruling can be no larger than the maximum possible dimensionof an isotropic subspace. For a 2 n -dimensional hypersurface, the maximum possible dimensionof an isotropic subspace is n , and so the maximal rank of a ruling is n , and a nondegeneratehypersurface admitting such a ruling necessarily has split signature second fundamental form. Theexamples constructed here are maximally ruled in the sense that they carry rulings with the maximalpossible rank. It would be interesting to know if there are maximally ruled equiaffine mean curvaturehypersurfaces that are not equivalent to these examples.The hypersurfaces constructed are realized as level sets, and so it is necessary to record, inSection 2, some facts and formulas related to the equiaffine geometry of level sets that are neededfor the proofs, as this material is hard to find in the literature. A formula for the equiaffine normalof a level set of F is given in J. Hao and H. Shima’s [11] under the condition that the Hessian F ij of F be nondegenerate, which is too restrictive in applications. In particular, in the exampleshere, F ij has corank one everywhere. Fortunately, the formulas in [11] remain valid under lessrestrictive hypotheses, provided they are properly interpreted. Let U ij be the adjugate tensor ofthe Hessian F ij and let U ( F ) = U ij F i F j , where F i is the differential of f . Then the level set of F containing the regular point p is nondegenerate at p if and only if U ( F ) is not zero at p . Thisobservation is due to R. Reilly in [17]. Since the claim and the paper [17] seem little known, theproof is reviewed here. More generally, there are derived in terms of U ij and U ( F ) formulas forthe equiaffine normal and equiaffine metric of a level set valid under the hypothesis that U ( F ) notvanish. Actually, slightly more is obtained, in that there is constructed from F a vector field W that along each level set of F agrees with the equiaffine normal of the level set. The differentialoperator associating wtih F the vector field W is invariant under orientation-preserving externalreparameterizations (meaning replacing F by Ψ ◦ F for an orientation-preserving diffeomorphismΨ), equivariant (in a particular sense) with respect to the action on functions of the group of affinetransformation, and invariant with respect to the action on functions of the group of equiaffinetransformations. A further condition is necessary to determine this operator uniquely. See section2 for further discussion.The examples constructed in Section 3 are the level sets of a function F satisfying that its Hessianhas corank one and U ( F ) is equal to a constant everywhere. The complete description of thesolutions of these equations seems an interesting problem on its own. The examples described hereshow that examples abound, although the equiaffine metrics of the examples have split signature.This should be compared with the situation for convex hypersurfaces with equiaffine mean curvaturezero. In [19], N. Trudinger and X. J. Wang have shown the validity of the affine Bernstein conjecture,that a locally uniformly convex hypersurface in R that is complete in the equiaffine metric and has QUIAFFINE GEOMETRY OF LEVEL SETS 3 equiaffine mean curvature zero is an elliptic paraboloid. Moreover, in [20] they conjecture that thesame result should hold for hypersurfaces in R n for n ≤ Equiaffine geometry of level sets
Let b ∇ be a torsion-free affine connection on the ( n + 1)-dimensional manifold M . The secondfundamental form of a co-orientable immersed hypersurface Σ in M with respect to b ∇ is the normalbundle valued symmetric covariant two-tensor on Σ equal, when evaluated on vector fields X and Y tangent to Σ, to the projection of b ∇ X Y onto the normal bundle of Σ. The hypersurface Σ is nondegenerate (at a point p ∈ Σ ) if its second fundamental form is nondegenerate everywhere (at p ). A vector field N transverse to Σ determines a splitting of the pullback of T M over Σ as thedirect sum of T Σ and the span of N . Via this splitting, the connection b ∇ induces on Σ a connection ∇ , while via N , the second fundamental form is represented by a symmetric covariant two tensor h on Σ. In particular, Σ is nondegenerate if and only if h is nondegenerate, and this conditiondoes not depend on the choice of the transversal N . For vector fields X and Y tangent to Σ, theconnection ∇ , the tensor h , the shape operator S ∈ Γ(End( T Σ)), and the connection one-form τ ∈ Γ( T ∗ Σ) are determined by the relations b ∇ X Y = ∇ X Y + h ( X, Y ) N, b ∇ X N = − S ( X ) + τ ( X ) N, (2.1)where here, as in what follows, notation indicating the restriction to Σ, the immersion, the pullbackof T M , etc. is omitted to improve readability, and Γ( E ) is the space of smooth sections of the vectorbundle E . By (2.1), for X tangent to Σ, and any volume form Ψ on M , ∇ X ( ι ( N )Ψ) = ι ( N ) b ∇ X Ψ + τ ( X ) ι ( N )Ψ . (2.2)By (2.2), if b ∇ Ψ = 0, then, along Σ, τ is determined by τ = ( ι ( N )Ψ) − ∇ ( ι ( N )Ψ). The meancurvature of Σ with respect to N and b ∇ is n − tr S .2.1. Nondegeneracy of level sets.
Let M = R n +1 and let b ∇ be the standard flat affine connectionon R n +1 . Fix a b ∇ -parallel volume form Ψ. The group Aff ( n + 1 , R ) of affine transformations of R n +1 comprises the automorphisms of b ∇ . Elements of its subgroup preserving Ψ are called unimodular or equiaffine . Let Ω ⊂ R n +1 be an open domain. For F ∈ C ∞ (Ω) let F i ...i k = b ∇ i . . . b ∇ i k − dF i k ,and let F ij = (Hess F ) ij = b ∇ i dF j be the Hessian of F . Here, as generally in what follows,the abstract index and summation conventions are employed (the reader unfamiliar with these DANIEL J. F. FOX conventions can consult chapter 2 of [15]). As det Hess F and the tensor square Ψ are 2-densities,it makes sense to define the Hessian determinant H ( F ) of the at least twice differentiable function F by det Hess F = H ( F )Ψ . By affine coordinates are meant smooth functions x , . . . , x n +1 suchthat the differentials dx , . . . , dx n +1 are linearly independent and constitute a b ∇ -parallel coframe;these coordinates are equiaffine if, moreover, Ψ = dx ∧ · · · ∧ dx n +1 . In equiaffine coordinates, H ( F ) = det ∂ F∂x i ∂x j .Formally the adjugate tensor of a symmetric covariant two-tensor F ij is a 2-density valuedsymmetric contravariant two-tensor ¯ U ij satisfying ¯ U ip F pj = (det Hess F ) δ j i . Here, instead, thesymmetric contravariant two-tensor U ij defined by tensoring ¯ U ij with Ψ − will be called the adju-gate tensor of F ij . Its characteristic property is U ip F pj = H ( F ) δ j i . Where H ( F ) is nonzero, F ij isa pseudo-Riemannian metric with inverse symmetric bivector F ij , and there hold F ij = H ( F ) − U ij and H ( F ) − U ip F p = F ij F j . Like U ij , the vector field N i = U ip F p and the function U ( F ) = N i F i = U ij F i F j are defined even when F ij degenerates. The adjugate transformation of an endomorphismof an r -dimensional vector space has rank r , 1, or 0 as the original transformation has rank r , r − r −
1. Consequently, if H ( F ) vanishes then U ij has rank 1 or 0 as F ij has rank n orrank less than n . When U ij has rank 1, something more precise can be said; see Lemma 2.5 below.Applying the Cauchy determinantal identity (Equation (19) in [2]), (cid:12)(cid:12)(cid:12)(cid:12) A bc t d (cid:12)(cid:12)(cid:12)(cid:12) = (det A ) d − c t (adj A ) b, (2.3)where A is a matrix with adjugate matrix adj A , b and c are column vectors, and d ∈ R , yields theidentity (cid:12)(cid:12) A + bc t (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) A + bc t b (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) A b − c t (cid:12)(cid:12)(cid:12)(cid:12) = | A | + c t (adj A ) b. (2.4)Applying (2.3) and (2.4) to tensors yields U ( F ) = U ij F i F j = − (cid:12)(cid:12)(cid:12)(cid:12) F ij F i F j (cid:12)(cid:12)(cid:12)(cid:12) , (2.5) det( F ij + qF i F j ) = ( H ( F ) + q U ( F ))Ψ . (2.6)for any smooth function q . Notation is abused in (2.5) in that a covariant tensor is apparentlyidentified with an endomorphism; if F i and F ij are interpreted as the components of dF and b ∇ dF with respect to equiaffine coordinates the formulas make rigorous sense, and this justifies their usegenerally. Alternatively the matricial notation can be understood as an abstract notational devicelike the abstract index conventions. Lemma 2.1.
For F ∈ C ∞ ( R n +1 ) and g ∈ Aff ( n + 1 , R ) define ( g · F )( x ) = F ( g − x ) . Let ℓ : Aff ( n + 1 , R ) → GL ( n + 1 , R ) be the projection onto the linear part. Then g · H ( F ) = det ℓ ( g ) H ( g · F ) , g · U ( F ) = det ℓ ( g ) U ( g · F ) , (2.7) Proof.
There hold ℓ ( g ) i j ( g · F ) j ( x ) = F i ( g − x ) and ℓ ( g ) i a ℓ ( g ) j b ( g · F ) ab ( x ) = F ij ( g − x ). Takingthe determinant of the last yields the first equality of (2.7), while substituting both into (2.5) yieldsthe second identity of (2.7). (cid:3) Remark 2.1.
Let Φ : R n +1 → R n +1 be a linear fractional transformation, so that Φ( x ) i = ( c p x p + d ) − ( A q i x q + b i ). Then Φ j i = ∂∂x j Φ i and Φ ij k = ∂ ∂x i ∂x j Φ k satisfy Φ jk i = − Φ j i c k − Φ k i c j . By QUIAFFINE GEOMETRY OF LEVEL SETS 5 (2.5), − U ( F ◦ Φ) = (cid:12)(cid:12)(cid:12)(cid:12) F pq Φ i p Φ j q + F p Φ ij p F p Φ i p F q Φ j q (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) F pq Φ i p Φ j q − F p Φ ( i p c j ) F p Φ i p F q Φ j q (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) F pq Φ i p Φ j q F p Φ i p F q Φ j q (cid:12)(cid:12)(cid:12)(cid:12) = − (det T Φ) U ( F ) ◦ Φ . (2.8)The identity (2.8) shows that U ( F ) transforms equivariantly under precomposition with projectivetransformations, and yields the second identity of (2.7) as a special case. This projective covarianceof U ( F ) is a reason for paying special attention to this quantity. ♦ A point p ∈ Ω is a regular point of F if dF is not zero at p . Because the set of regular pointsin Ω is open, the level sets of the restriction of F to a sufficiently small neighborhood of a regularpoint of F are smoothly immersed submanifolds. It makes sense to say that the level set of F containing p is degenerate or nondegenerate at p because the part of this level set contained in asufficiently small neighborhood of p is smoothly immersed. The level set of F containing p means { x ∈ Ω : F ( x ) = F ( p ) } .A Euclidean metric on the flat equiaffine space ( R n +1 , Ψ , b ∇ ) means a b ∇ -parallel Riemannianmetric the volume form of which equals Ψ. Lemma 2.2 gives a geometric interpretation of U ( F ) interms of the Gauss-Kronecker curvature K of a level set of F with respect to a Euclidean metric. Lemma 2.2.
Let δ ij and δ ij be a Euclidean metric and its inverse on R n +1 . Let F be a C ∞ functiondefined on an open subset Ω ⊂ R n +1 . At a regular point p ∈ Ω of F , the Gauss-Kronecker curvature K with respect to the Euclidean unit normal vector −| dF | − δ δ ip F p of the smoothly immersed levelset of F passing through p satisfies U ( F ) = K | dF | n +2 δ . (2.9) Proof.
Write E i = −| dF | − δ δ ip F p and E i = E p δ ip = −| dF | − δ F i . The tensorΠ ij = | dF | − δ (cid:0) F ij − E p F p ( i E j ) + E p E q F pq E i E j (cid:1) (2.10)satisfies E i Π ij = 0 and its restriction to the tangent space to a level set of F is the representativeof the second fundamental form of the level set with respect to E i . The tensor Λ ij = Π ij + E i E j is nondegenerate and its determinant satisfies (det Λ) / Ψ = (det Π) / ( ι ( E )Ψ) = K where det Πmeans the determinant of the restriction of Π to a tangent space of a level set of F , and K is theGauss-Kronecker curvature with respect to E ( K is defined by the preceding relation). Elementaryoperations with determinants coupled with (2.10) and (2.5) yield (abusing notation as in (2.5)) − K = (cid:12)(cid:12)(cid:12)(cid:12) Λ ij E j − (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) Π ij E j E i − (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) Π ij E j E i (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) | dF | − δ F ij E j E i (cid:12)(cid:12)(cid:12)(cid:12) = −| dF | − n − δ U ( F ) , (2.11)from which (2.9) follows. (cid:3) Corollary 2.3 (Corollary of Lemma 2.4) . Let F be a C ∞ function defined on an open subset Ω ⊂ R n +1 and satisfying U ( F ) = 0 on Ω . If r ∈ R is a regular value of F and the level set Σ r ( F, Ω) = { x ∈ Ω : F ( x ) = r } is nonempty, then Σ r ( F, Ω) is a smoothly immersed hypersurfaceof Gauss-Kronecker curvature zero.Proof. Since r is regular, | dF | δ does not vanish on Σ r ( F, Ω) and the claim follows from (2.9). (cid:3)
Lemma 2.4.
Let F be a C ∞ smooth function on a nonempty open subset Ω ⊂ R n +1 . For p ∈ Ω the following are equivalent.(1) U ( F ) is not zero at p .(2) p is a regular point of F and the Gauss-Kronecker curvature of F at p with respect to anyEuclidean metric is nonzero. DANIEL J. F. FOX (3) p is a regular point of F and the level set of F containing p is nondegenerate at p .(4) p is a regular point of F and the restriction of the Hessian of F to the level set of F containing p is nondegenerate at p .Proof. The equivalence of (1) and (2) is immediate from (2.9). The equivalence of (2) and (3)results from the observations that the shape operator of a Riemannian metric is invertible if andonly if the representative of the second fundamental form associated with a unit normal vector fieldis nondegenerate and that the second fundamental form is nondegenerate if and only if its repre-sentative with respect to any transversal is nondegenerate. There remains to show the equivalenceof (3) and (4). If p is a regular point then the level set Σ containing p of the restriction of F to asmall neighborhood of p is smoothly immersed and there is a vector field V transverse to Σ near p and satisfying F i V i = 0 in a neighborhood in Σ of p . Let h be the representative of the secondfundamental form of Σ associated with V and let X and Y be vector fields tangent to Σ near p .Then, along Σ, ( b ∇ X dF )( Y ) = − dF ( b ∇ X Y ) = − h ( X, Y ) dF ( V ). Since dF ( V ) = 0 at p , it followsthat h is nondegenerate if and only if the restriction of F ij to Σ is nondegenerate at p . (cid:3) Remark 2.2.
The main point of Lemma 2.4 is the equivalence of (1) and (2). This was proved byR. Reilly as Proposition 4 of [17]. ♦ Remark 2.3.
The deduction of the equivalence of (1) and (3) of Lemma 2.4 via the identity(2.9) relating U ( F ) to the Gauss-Kronecker curvature is not completely satisfying because of itsuse of an apparently extraneous Euclidean structure. Here is given a direct proof that (3) implies(1) using only the equivalence of (3) and (4) and making no use of any metric structure. (Analternative proof that (1) implies (3) is given in the proof of Lemma 2.5 below.) It is claimed thatif U ( F ) vanishes at a regular point p then there is a vector field X tangent to Σ at p and suchthat X p F ip = 0. By the equivalence of (3) and (4) of Lemma 2.4, this suffices to show that Σ isdegenerate at p . If U ( F ) vanishes at p and N i does not vanish at p , then for any vector field Y tangent to Σ at p there holds 0 = H ( F ) dF ( Y ) = ( b ∇ Y dF )( N ), which shows that F ij degenerates at p . If U ij vanishes at p then the rank of F ij at p is at most n −
1, so its restriction to Σ is degenerateat p . Finally, if N vanishes at p and U ij has rank 1 at p then there are a vector field X and asmooth function c defined in a neighborhood of p such that at p there holds U ij = cX i X j . If p is aregular point, then, since 0 = N i = U ij F j = cX i X p F p , X is tangent to Σ at p . On the other hand,since 0 = U ip F pj = cX i X p F ip at p , there holds X p F ip = 0 at p . This shows that the negation of(1) implies the negation of (3); precisely, if U ( F ) vanishes at p then Σ is degenerate at p or p is acritical point, and the latter possibility can occur only if N vanishes at p . ♦ Lemma 2.5.
Let F be a C ∞ function defined on an open subset of R n +1 and let Ω be a connectedcomponent with nonempty interior of the set where U ( F ) does not vanish. Define Π ij = F ij − U ( F ) − H ( F ) F i F j . Suppose Σ r ( F, Ω) = { x ∈ Ω : F ( x ) = r } is nonempty. Then:(1) N i = U ip F p does not vanish on Ω and each nonempty level set Σ r ( F, Ω) is a smoothlyimmersed nondegenerate hypersurface co-oriented by N . The representative of the secondfundamental form of Σ r ( F, Ω) with respect to the transversal N is the restriction of h ij = − U ( F ) − Π ij = − U ( F ) − (cid:0) F ij − H ( F ) U ( F ) − F i F j (cid:1) , (2.12) and the signature of h ij is constant on Ω .(2) For any nonvanishing q ∈ C ∞ (Ω) the tensor m ij = Π ij + q U ( F ) − F i F j = F ij − U ( F ) − H ( F ) F i F j + q U ( F ) − F i F j , (2.13) is nondegenerate and det m = q Ψ . Let m ij be the inverse of m ij defined by m ip m pj = δ j i . The tensor Π ij = m ij − q − U ( F ) − N i N j does not depend on the choice of q . Since QUIAFFINE GEOMETRY OF LEVEL SETS 7 Π ip F p = 0 , it makes sense to speak of the restriction of Π ij to Σ r ( F, Ω) , and this restrictionis the inverse to the restriction of Π ij to Σ r ( F, Ω) .(3) There holds U ij − U ( F ) − N i N j = H ( F )Π ij . At a point p ∈ Ω where H ( F ) vanishes, U ij = U ( F ) − N i N j and F ij has rank n .(4) If ∆ = { x ∈ Ω : H ( F )( x ) = 0 } has nonempty interior, then H ( F ) − ( U ij − U ( F ) − N i N j ) extends smoothly to the closure ¯∆ , where it equals Π ij .(5) Along Σ r ( F, Ω) , there holds | U ( F ) | ( n +1) / | vol h | = | ι ( N )Ψ | where | vol h | is the volume densityinduced on Σ r ( F, Ω) by the metric h ij of (2.12) .Proof. Since dF does not vanish on Ω, if Σ r ( F, Ω) is nonempty then it is a smoothly immersedhypersurface, and since N i F i = U ( F ) also does not vanish on Ω, N i is transverse to Σ r ( F, Ω). By(2.1), for vector fields X and Y tangent to Σ r ( F, Ω) there holds ( b ∇ X dF )( Y ) = − U ( F ) h ( X, Y ).Together with N p F ip = H ( F ) F i this shows that the tensor (2.12) satisfies N p h ip = 0 and thatits restriction to Σ r ( F, Ω) is the representative of the second fundamental form determined by N i .Applying (2.6) yields det m = q Ψ , so m ij is nondegenerate. Because Ω is connected and, in thespace of symmetric bilinear forms on a vector space, a connected component of the subspace ofnondegenerate forms comprises forms of a fixed signature, the smooth nondegenerate form m ij cannot change signature on Ω. Since m ij = − U ( F ) h ij + U ( F ) − F i F j , N p m ip = qF i annihilates thetangent space to Σ r ( F, Ω), and N i N j m ij = U ( F ) q has constant sign on Ω, the tensor h ij cannotchange signature on Ω.Let ˜ m ij be defined as m ij in (2.13), but with the nonvanishing function ˜ q ∈ C ∞ (Ω) in place of q . Then ˜ m ij − m ij = ( ˜ m ip − m ip ) m pq m qj = ˜ m ip ( ˜ m pq + ( q − ˜ q ) U ( F ) − F p F q ) m qj − m ij = (˜ q − − q − ) U ( F ) − N i N j , (2.14)from which it follows that the tensor Π ij does not depend on q .The tensor Π ij satisfies N j Π ij = 0, so N j m ij = qF i . Consequently, N i = qm ip F p , and soΠ ij F j = 0. Hence it makes sense to speak of the restriction of Π ij to Σ r ( F, Ω) . Since Π ij F j = 0,Π ip Π pj = Π ip m pj = ( m ip − q − U ( F ) − N i N p ) m pj = δ j i − U ( F ) − F j N i , (2.15)which shows that the restriction of Π ij to Σ r ( F, Ω) is the symmetric tensor inverse to the restrictionto Σ r ( F, Ω) of Π ij . Because ( U ip − U ( F ) − N i N p ) F p = 0,( U ip − U ( F ) − N i N p ) m pj = ( U ip − U ( F ) − N i N p ) F pj = H ( F )( δ j i − U ( F ) − F j N i )(2.16)Raising the index j in (2.16) and substituting N i = qm ip F p into the result yields (3). By (3), where H ( F ) vanishes there holds U ij = U ( F ) − N i N j . Since Π ij is smooth on Ω, it follows from (3) that,whenever the subset ∆ is nonempty, the tensor H ( F ) − ( U ij − U ( F ) − N i N j ) extends smoothly tothe closure ¯∆, where it equals Π ij .Let m ij be the tensor defined in (2.13) with q = U ( F ) − , so that, by (2), det m = Ψ and m ij F i F j = U ( F ). By the definition of the volume densities | vol h | and | vol m | of the metrics h and m , for any X , . . . , X n tangent to Σ and u = | U ( F ) | − / there holds | ( ι ( N )Ψ)( X , . . . , X n ) | = u − ( n +1) | Ψ ( uN, uX , . . . , uX n ) | = u − ( n +1) | vol m ( uN, uX , . . . , uX n ) | = | U ( F ) | ( n +1) / | vol h ( X , . . . , X n ) | . (2.17)This shows (5). (cid:3) Remark 2.4.
A consequence of Lemma 2.5 is that when H ( F ) = 0 on all of Ω the tensor Π ij isstill defined. This means that formulas obtained assuming H ( F ) = 0 and involving H ( F ) − ( U ij − U ( F ) − N i N j ) continue to make sense where H ( F ) vanishes, provided that U ( F ) does not vanish. ♦ DANIEL J. F. FOX
Example 2.5.
Allowing H ( F ) to vanish is useful because in interesting examples it occurs thatthe level sets of F are nondegenerate although H ( F ) is identically zero. For example, along thehelicoid defined by the vanishing of F ( u, x, y ) = x sin u + y cos u , the Hessian of F degenerates, butit follows from Lemma 2.5 that this level set is a nondegenerate hypersurface, because U ( F ) = − ♦ Example 2.6 shows that it can occur that at a regular point p of F the Hessian of F has corankone and U ( F ) vanishes. Lemma 2.4 implies that in this case the level set of F containing p isdegenerate at p . Example 2.6.
It can occur that U ij has rank 1 (so Hess F has corank 1 and H ( F ) = 0) and U ( F )vanishes. The following example is based on the construction in section 7 of [10]. Let k ≥ a , b , and c be linearly independent homogeneous degree k polynomials of the variables x and x . Then P = a ( x , x ) x + b ( x , x ) x + c ( x , x ) x ∈ Pol k +2 ( R ) is irreducible and not affinelyequivalent to a polynomial of less than five variables but solves H ( P ) = 0, for the kernel of theHessian of P contains the vector field V = ( b c − b c ) ∂ + ( c a − c a ) ∂ + ( a b − a b ) ∂ , (2.18)where subscripts indicate first partial derivatives. A concrete example is P = x x + x x x + x x .See [5] for many related examples. Straightforward calculations show that U ij = V i V j , so that U ( P ) = dP ( V ) . However, dP ( V ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a a b b b b c c c (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 . (2.19)is the determinant of the 1-jet of the mapping from R to R having components a , b , and c , andthis vanishes because, by the Euler identity, ka = x a + x a and similarly for b and c . Hence U ( P ) = 0 although U ij has rank 1.Since the Euclidean norm | dP | δ does not vanish along a regular level set Σ of P , by Corollary 2.3the Gauss-Kronecker curvature of such a level set is zero. That is Σ is a developable hypersurface. ♦ Example 2.11, at the end of Section 2.2, illustrates some issues related to connected componentsof level sets and the signatures of their second fundamental forms.2.2.
Equiaffine normal vector field.
This section describes the equiaffine normal, and its asso-ciated tensors, of a level set of a function F ∈ C ∞ (Ω), where Ω ⊂ R n +1 is an open set, satisfyingthat U ( F ) does not vanish on Ω. Lemma 2.6.
Let F be a C ∞ function defined on an open subset of R n +1 . Let Ω be a connectedcomponent with nonempty interior of the region on which U ( F ) does not vanish. Then a nonemptylevel set Σ r ( F, Ω) is nondegenerate and the vector field W defined by W i = −| U ( F ) | / ( n +2) U ( F ) − N i − k ip µ p , (2.20) where k ij = | U ( F ) | / ( n +2) Π ij , Π ij is defined in (2) of Lemma 2.5, and µ = ( n + 2) − d log U ( F ) , istransverse to Σ r ( F, Ω) . The restriction to Σ r ( F, Ω) of the tensor k ij = | U ( F ) | − / ( n +2) Π ij = | U ( F ) | − / ( n +2) ( F ij − U ( F ) − H ( F ) F i F j ) , (2.21) represents the second fundamental form of Σ r ( F, Ω) with respect to W .Proof. The nondegeneracy of the level sets of F is the conclusion of Lemma 2.5. By (1) of Lemma2.5, k ij represents the second fundamental form of Σ r ( F, Ω) with respect to W . (cid:3) QUIAFFINE GEOMETRY OF LEVEL SETS 9
The transversal W i is the equiaffine normal vector field associated with F . This terminology isjustified by Theorem 2.15 that shows that, along a level set of F , W i agrees with the equiaffinenormal of the level set as usually defined. By Lemma 2.5, the tensor k ij of (2.21) has rank n wherever U ( F ) is not zero, and, since N j k ij = 0, by (2.12) and (2.33), the restriction of k ij toa level set of F is the representative of the second fundamental form determined by W ; it is the equiaffine metric of the level set. Note that, although the signature of k ij need not be the same ondistinct connected components of a level set of F , on those contained within a connected componentof the complement of the zero set of U ( F ), it does not change, by (1) of Lemma 2.5. Remark 2.7.
After translating notation, the formula (2.33) for the equiaffine normal can beidentified, up to sign, with that obtained, under the assumption H ( F ) = 0, by J. Hao and H. Shimain Theorem 1 of [11]. Precisely, in the notation used here the formula of [11] can be writtensgn( UH ) | U | / ( n +2) (cid:0) U − H F ip F p + F ip µ p − U − H F pq F p µ q F ia F a (cid:1) (2.22)where H = H ( F ), U = U ( F ), and F ij = H ( F ) − U ij is the symmetric tensor satisfying F ip F pj = δ j i (which exists because H ( F ) = 0). It is not obvious that the expression (2.22) continues to makesense when H ( F ) = 0 but U ( F ) = 0. That this is so follows from Lemma 2.5; precisely, (2.22)equals − sgn( U ( F ) H ( F )) W i . ♦ Let I ⊂ R be a connected open interval and let ψ : I → R be a C diffeomorphism. The levelsets of F and ψ ◦ F are the same, just differently parameterized, in the sense that for r ∈ I thereholds Σ r ( F, F − ( I ) ∩ Ω) = Σ ψ ( r ) ( ψ ◦ F, Ω), and so objects depending only on the geometry of thelevel sets need to transform well under such external reparameterizations.
Lemma 2.7.
For an open domain Ω ⊂ R n +1 , define U (Ω) = { F ∈ C ∞ (Ω) : U ( F )( x ) = 0 for all x ∈ Ω } . If F ∈ U (Ω) , I ⊂ F (Ω) is a connected open subinterval, and ψ : I → R is a smooth diffeomor-phism onto its image then ψ ◦ F ∈ U (Ω) .Proof. There holds ( ψ ◦ F ) ij = ˙ ψF ij + ¨ ψF i F j . (2.23)From (2.5) there results U ( ψ ◦ F ) = − (cid:12)(cid:12)(cid:12)(cid:12) ( ψ ◦ F ) ij ( ψ ◦ F ) i ( ψ ◦ F ) j (cid:12)(cid:12)(cid:12)(cid:12) = − (cid:12)(cid:12)(cid:12)(cid:12) ˙ ψF ij + ¨ ψF i F j ˙ ψF i ˙ ψF j (cid:12)(cid:12)(cid:12)(cid:12) = − ˙ ψ n +2 (cid:12)(cid:12)(cid:12)(cid:12) F ij F i F j (cid:12)(cid:12)(cid:12)(cid:12) = ˙ ψ n +2 U ( F ) . (2.24)By (2.24), the assumption that U ( F ) not vanish on Ω is preserved by external reparameterization. (cid:3) Remark 2.8.
Computing the determinant of (2.23) using (2.4) yields H ( ψ ◦ F ) = ˙ ψ n +1 ( H ( F ) + ( ¨ ψ/ ˙ ψ ) U ( F )) . (2.25)By Lemma 2.7, U is preserved by external reparameterizations whereas, by (2.25), the analogousstatement is not true for the subset comprising F ∈ C ∞ (Ω) with nonvanishing H ( F ). This obser-vation gives another reason for the prominent role played by U ( F ). ♦ An operator A : U (Ω) → Γ( T Ω) is invariant under (smooth) external reparameterizations if forevery F ∈ U (Ω), every connected open subset I ⊂ F (Ω), and every smooth ψ : I → R mapping I diffeomorphically onto its image there holds A ( ψ ◦ F ) i = sgn( ˙ ψ ) A ( F ) i . (2.26) Let L g be the operator of left multiplication by g ∈ Aff ( n + 1 , R ) on R n +1 , so that g · F = F ◦ L g − .Then A is equiaffinely invariant if for every g ∈ Aff ( n + 1 , R ) there holds L ∗ g − ( A ( F )) i = | det ℓ ( g ) | / ( n +2) A ( g · F ) i (2.27)at every point x ∈ Ω such that g − x ∈ Ω. Note that (2.27) actually demands that A be affinelycovariant in a particular way. There could be considered a rule like (2.27) with an arbitrarycharacter χ : Aff ( n + 1 , R ) → R × in place of | det ℓ ( g ) | / ( n +2) . The particular choice of character isexplained as follows. The interest here is in operators A ( F ) that are transverse in the sense thatthey satisfy additionally the condition that dF i A ( F ) i is nonvanishing on Ω for F ∈ U (Ω). Thisguarantees that at each p ∈ Ω the vector field A ( F ) i is transverse to the level set Σ of F containing p . On Σ there are two natural volume densities determined by A ( F ). These are | ι ( A ( F ))Ψ | andthe volume density of the representative of the second fundamental form of Σ corresponding to A ( F ). The choice of character in (2.27) is determined by requiring that these two volume densitiesrescale in the same way when F is replaced by g · F . Precisely, suppose L ∗ g − ( A ( F )) = χ ( g ) A ( g · F ).Then | ι ( A ( g · f )Ψ | = | det ℓ ( g ) | − | χ ( g ) || ι ( A ( F ))Ψ | and the volume densities | vol ˜ h | and | vol h | of therepresentatives ˜ h and h of the second fundamental form of Σ corresponding respectively to A ( g · F )and A ( F ) are related by | vol ˜ h | = | χ ( g ) | − n/ | vol h | . Hence | ι ( A ( g · f ))Ψ | / | vol ˜ h | = | ι ( A ( F )Ψ | / | vol h | ifand only if | χ ( g ) | = | det ℓ ( g ) | / ( n +2) . Theorem 2.8.
For an open domain Ω ⊂ R n +1 , the transverse operators associating with F ∈ U (Ω) the vector fields W i (where W is defined in (2.20) ) and −| U ( F ) | / ( n +2) U ( F ) − N i on Ω are invariantunder external reparameterizations and equiaffinely invariant.Proof. For readability there are written ˙ ψ , ¨ ψ , etc. instead of ˙ ψ ( F ), ¨ ψ ( F ), etc. The objects derivedfrom ψ ◦ F in the same manner as those derived from F are decorated with a ˜. For example (2.23)is written ˜ F ij = ˙ ψF ij + ¨ ψF i F j . By (2.23) and (2.25), the tensor Π ij = F ij − H ( F ) U ( F ) − F i F j defined in Lemma 2.5 transforms by ˜Π ij = ˙ ψ Π ij . (2.28)By Lemma 2.5, the restriction of ˜Π ij to a level set of ( ψ ◦ F ) is the inverse of the restriction of˜Π ij = ˙ ψ Π ij to this level set. Since this level set is also a level set of F and the restriction to it of˜Π ij is the inverse of the restriction of Π ij it follows that˜Π ij = ˙ ψ − Π ij . (2.29)By (2.24) and (2.28), the tensor k ij = | U ( F ) | − / ( n +2) Π ij defined in (2.21) transforms as ˜ k ij =sgn( ˙ ψ ) k ij . Since the tensor k ij = | U ( F ) | / ( n +2) Π ij satisfies k ip F p = 0 it makes sense to speak ofthe restriction of k ij to a level set of F , and, since k ip k pj = δ j i − U ( F ) − F j N i , this restriction isthe inverse of the restriction of k ij . By (2.15), (2.24), (2.28), and (2.29) U ( F ) − F i N j = δ i j − Π jp Π ip = δ i j − ˜Π jp ˜Π ip = U (( ψ ◦ F )) − ˜ F i ˜ N j = ˙ ψ − n − U ( F ) − F i ˜ N j . (2.30)Since U ( F ) − F i does not vanish, this shows˜ N i = ˜ U ip ( ψ ◦ F ) p = ˙ ψ n +1 N i . (2.31)Together (2.24) and (2.31) show that −| U ( F ) | / ( n +2) U ( F ) − N i satisfies (2.26). It follows from(2.24) that the one-form µ = ( n + 2) − d log U ( F ) transforms by µ ( ψ ◦ F ) i = µ i + ( ¨ ψ/ ˙ ψ ) F i . (2.32) QUIAFFINE GEOMETRY OF LEVEL SETS 11
The vector field Z i = U ( F )Π ip µ p satisfies Z p F p = 0 and Z p Π ip = U ( F ) µ i − N p µ p F i . By (2.24),(2.29), and (2.32), ˜ Z i = ˙ ψ n +1 Z i . By (2.7), (2.25), (2.24), and (2.31), the vector field W i = − sgn( U ( F )) | U ( F ) | − ( n +1) / ( n +2) (cid:0) N i + Z i (cid:1) = −| U ( F ) | / ( n +2) U ( F ) − N i − k ip µ p , (2.33)transforms by ˜ W i = sgn( ˙ ψ ) W i . This shows that W is invariant under external reparameterizations.There remains to check (2.27). Now, write ˜ F ij = ( g · F ) ij for g ∈ Aff ( n + 1 , R ). More generally,a tensor or scalar constructed from g · F is indicated using the same notation as that indicating thecorresponding tensor or scalar constructed from F , but decorated with a ˜. By construction L ∗ g ( d ( g · F )) = dF and L ∗ g (Hess( g · F )) = Hess F , where L g indicates the action by left multiplication of g ∈ Aff ( n + 1 , R ) on R n +1 . It is convenient to write ℓ i j = ℓ ( g ) i j and ¯ ℓ i j for the inverse endomorphism(so ¯ ℓ p j ℓ i p = δ i j ). Then ( g · F ) i ( x ) = ¯ ℓ i p F p ( g − x ) and ˜ F ( x ) ij = ¯ ℓ i p ¯ ℓ j q F pq ( g − x ). With (2.7)it follows that ˜ U ij ( x ) = (det ℓ ( g )) − ℓ a i ℓ b j U ab ( g − x ) and so ˜ N i ( x ) = (det ℓ ( g )) − ℓ a i N a ( g − x ).Equivalently, (det ℓ ( g )) − L ∗ g − ( N ) i = ˜ N i . This suffices to show that −| U ( F ) | / ( n +2) U ( F ) − N i satisfies (2.27). From (2.7) it follows that the one-form ( n +2) µ ( F ) i = d i log U ( F ) satisfies µ ( g · F ) = L ∗ g − µ ( F ). Similarly, ˜Π ij ( x ) = ¯ ℓ i p ¯ ℓ j q Π pq ( g − x ). From (2.15) and (2.7) it follows that ˜Π ij = ℓ p i ℓ q j Π pq ( g − x ). It follows that ˜ Z i ( x ) = (det ℓ ( g )) − ℓ a i Z a ( g − x ). Assembling the precedingshows W satisfies (2.27).Alternatively, Theorem 2.15 below shows that along a level set of F the transversal W i agrees withthe equiaffine normal of the level set, as usually defined, and the affine covariance (2.27) is true byconstruction for the usual affine normal, so there is really no need to check it directly. (Note that thesame argument does not apply to show the equiaffine invariance of −| U ( F ) | / ( n +2) U ( F ) − N i .) (cid:3) Example 2.9.
Here it is shown how to recover from (2.33) the usual formula (see, for example,equation (3 .
4) in [14]) for the equiaffine normal of a graph.Let V be an ( n + 1)-dimensional vector space equipped with the standard equiaffine structure( ∇ , Ψ), where Ψ is given by the determinant, and let V = W ⊕ L , where the subspace W hascodimension one. Let 0 = v ∈ V span L and let ν ∈ V ∗ satisfy ν ( v ) = 1 and ker ν = W . Equip W with the induced affine structure and the parallel volume form ω = ι ( v )Ψ and define the operator H on W with respect to this induced equiaffine structure. Given f ∈ C ∞ ( W ) define F = ν − f ◦ π ,where π : V → W is the projection along L . It is convenient to abuse notation by suppressing π and identifying f with its pullback f ◦ π to W . The graph { ( w, f ( w ) v ) ∈ W ⊕ L } of f along v is contained in the level set Σ = { x ∈ V : F ( x ) = 0 } . Lowercase Latin indices indicate tensorson V while uppercase Latin indices indicate tensors on W . Tensors on L are written as functions,without labels. Using block matrix notation in a formal way to indicate the decomposition oftensors corresponding to the splitting V = W ⊕ L , F i = (cid:0) − f I (cid:1) , F ij = (cid:18) − f IJ
00 0 (cid:19) , U ij = (cid:18) − n H ( f ) (cid:19) . (2.34)From (2.34) it is apparent that H ( F ) = 0, N = U ip F p = ( − n H ( f ) v , U ( F ) = ( − n H ( f ), and( n + 2) µ = d log H ( f ). Since, by Lemma 2.4, Σ is nondegenerate at p ∈ Σ if and only if U ( F ) isnonzero at p , it follows that Σ is nondegenerate at p if and only if H ( f ) is nonzero at the projectionof p onto W along L . In this case, let f IJ be the symmetric bivector on W inverse to the Hessian f IJ = ∇ I df J and let Z I be the pushforward of f IQ d Q log | H ( f ) | / ( n +2) via the immersion W → V given by w → w + f ( w ) v . Since µ ( N ) = 0 and h ij = ( − n H ( f ) F ij , it follows from (2.33) that theequiaffine normal of the graph of f along v has the expression W = −| H ( f ) | / ( n +2) ( v − Z ).After an equiaffine transformation, V , W , L , and the associated connections and volume forms canalways be put in the following standard form. Let V = R n +1 and Ψ = dx ∧ · · · ∧ dx n +1 , and regard R n as the subspace W = { x ∈ V : x n +1 = 0 } with the induced connection, also written ∇ , and thevolume form ω = dx ∧ · · · ∧ dx n . Here ν = dx n +1 and F ( x , . . . , x n +1 ) = x n +1 − f ( x , . . . , x n ). ♦ Once a co-orientation has been fixed, let ∇ , k , and S be the induced connection, metric, andshape operator of the level set of F determined by the co-oriented equiaffine normal. The pseudo-Riemannian metric k is the equiaffine metric, S is the equiaffine shape operator , and the equiaffinemean curvature H is the mean curvature with respect to W .Lemma 2.9 gives explicit expressions for S and H . The proof of Lemma 2.9 uses the identity b ∇ p U ip = 0. Differentiating U ip F pj = H ( F ) δ j i yields F pj b ∇ k U ip + U ip F jkp = H ( F ) k δ j i . Con-tracting this in i and k and using U pq F ipq = H ( F ) i gives F pj b ∇ q U qp = 0. Hence H ( F ) b ∇ p U ip = U ij F pj b ∇ q U qp = 0, so when H ( F ) = 0 there holds b ∇ p U ip = 0. However, in the proof this fact isneeded when H ( F ) = 0. In this generality the claim is a special case of a general identity for theNewton transforms of a symmetric endomorphism proved in Proposition 2 . Lemma 2.9.
Let F be a C ∞ function defined on an open subset of R n +1 . Let Ω be a connectedcomponent with nonempty interior of the region on which U ( F ) does not vanish. For r ∈ R , let Σ r ( F, Ω) = { x ∈ Ω : F ( x ) = r } . The equiaffine mean curvature H of Σ r ( F, Ω) is given by n H = b ∇ p ( k pq µ q ) + ( n + 2) | U ( F ) | / ( n +2) U ( F ) − ( H ( F ) − N p µ p )= b ∇ p ( k pq µ q ) + ( − n +1 ( n + 2) sgn( U ( F ))(det b ∇ ρ ) / Ψ , (2.35) where the equiaffine conormal one-form ρ i = −| U ( F ) | − / ( n +2) F i annihilates the tangent spaceto the level sets of F and satisfies ρ i W i = 1 .Proof. Differentiating (2.33) and using b ∇ i N j = H ( F ) δ i j + F p b ∇ i U jp yields b ∇ i W j = −| U ( F ) | / ( n +2) U ( F ) − (cid:16) H ( F ) δ i j − F p b ∇ i U jp + ( n + 1) µ i N j (cid:17) − b ∇ i ( k jp µ p ) . (2.36)Contracting (2.36) with ρ j and using d i U ( F ) = 2 H ( F ) F i + F p F q b ∇ i U pq (2.37)and k jp F ip = | U ( F ) | / ( n +2) Π jp (Π ip + U ( F ) − H ( F ) F j F p )= | U ( F ) | / ( n +2) ( δ j i − U ( F ) − F j N i )(2.38)yields ρ j b ∇ i W j = U ( F ) − ( N p µ p − H ( F )) F i . Consequently, S i j = − b ∇ i W j + U ( F ) − ( N p µ p − H ( F )) F i W j = b ∇ i ( k jp µ p ) + | U ( F ) | / ( n +2) U ( F ) − (cid:16) H ( F ) δ i j − F p b ∇ i U jp +( n + 1) µ i N j + ( H ( F ) − N p µ p ) ρ i W j (cid:1) , (2.39)satisfies S i p F p = 0, so it makes sense to speak of the restriction S I J of S i j to the tangent bundleof Σ r ( F, Ω), and S I J is the equiaffine shape operator of Σ r ( F, Ω). From (2.39) it follows that n H = S I I = S i i equals the first expression in (2.35). Differentiating ρ i shows b ∇ i ρ j = −| U ( F ) | − / ( n +2) F ij − µ i ρ j = − k ij + ( U ( F ) − H ( F ) F i − µ i ) ρ j . (2.40)Applying (2.4) to (2.40) yieldsdet b ∇ ρ = ( − n +1 | U ( F ) | − ( n +1) / ( n +2) ( H ( F ) − N p µ p )Ψ . (2.41)The second equality of (2.35) follows from (2.41). (cid:3) QUIAFFINE GEOMETRY OF LEVEL SETS 13
Theorem 2.8 shows that the conditions (2.26) and (2.27) do not characterize the transversal W i because Q i = −| U ( F ) | / ( n +2) U ( F ) − N i also satisfies these conditions. By the proof of Lemma 2.9,the one forms F p b ∇ i W p and F i are proportional. This is not true for Q i . Computations using (2.15)show that, for any one-form σ i , there holds F j b ∇ i ( k jp σ p ) = −| U ( F ) | / ( n +2) ( σ i − U ( F ) − N p µ p F i ).Consequently if M i = W i + k ip σ p , then F p F [ i b ∇ j ] M p = ρ [ i σ j ] , and so F p F [ i b ∇ j ] M p = 0 if and onlyif σ i is a multiple of F i . Since Q i = W i + k ip µ p , there holds F p F [ i b ∇ j ] Q p = 0 for all F ∈ U (Ω) ifand only if µ ∧ dF = 0 for all F ∈ U (Ω). It is straightforward to find F for which this is false, andso the condition that F p F [ i b ∇ j ] A ( F ) p = 0 selects W i in place of Q i .By (2.20), W and −| U ( F ) | / ( n +2) U ( F ) − N i coincide along a level set of F where U ( F ) is constant.By the following observation due to R. Reilly, given a level set of F there can always be found afunction G such that G and U ( G ) are constant along the given level. Lemma 2.10 (R. Reilly; Proposition 4 of [17]) . Let F be a C ∞ function defined on an opensubset of R n +1 . Let Ω be a connected component with nonempty interior of the region on which U ( F ) does not vanish. For r ∈ F (Ω) let G = | U ( F ) | − / ( n +2) ( F − r ) . Then | U ( G ) | = 1 along Σ r ( F, Ω) = Σ ( G, Ω) .Proof. Let q ∈ C ∞ (Ω) be nonvanishing and define G = q ( F − r ). By (2.5) and elementary deter-minantal computations − U ( G ) = (cid:12)(cid:12)(cid:12)(cid:12) qF ij + 2 F ( i q j ) + ( F − r ) q ij qF i + ( F − r ) q i qF j + ( F − r ) q j (cid:12)(cid:12)(cid:12)(cid:12) = q n +2 (cid:12)(cid:12)(cid:12)(cid:12) F ij − ( F − r ) q ( q − ) ij F i + ( F − r ) q − q i F j + ( F − r ) q − q j (cid:12)(cid:12)(cid:12)(cid:12) , (2.42)and, again by (2.5), along Σ r ( F, Ω) the last expression in (2.42) equals − q n +2 U ( F ). Hence, if q = | U ( F ) | − / ( n +2) , then U ( G ) = sgn( U ( F )). (cid:3) It is not generally possible to find a G having the same level sets as F and having U ( G ) constantalong each level set. The obstruction is identified in Lemma 2.11. On the other hand, when it ispossible, then there can be found G locally constant on the level sets of F and satisfying | U ( G ) | = 1.A version of Lemma 2.11, with the additional hypothesis that H ( F ) = 0 and slightly weakerconclusions was stated in [9]. Lemma 2.11.
Let F be a C ∞ function defined on an open subset of R n +1 . Let Ω be a connectedcomponent with nonempty interior of the region on which U ( F ) does not vanish. For r ∈ F (Ω) , let Σ r ( F, Ω) = { x ∈ Ω : F ( x ) = r } and let W be its equiaffine normal field (2.33) . The following areequivalent:(1) U ( F ) is locally constant on Σ r ( F, Ω) for all r ∈ F (Ω) .(2) The equiaffine conormal one-form ρ is closed, dρ = 0 .(3) µ ∧ ρ = 0 .(4) The local flow generated by W preserves the level sets of F .(5) ρ is constant along the local flow generated by W .(6) W ∧ N = 0 .(7) W = − sgn( U ( F )) | U ( F ) | − ( n +1) / ( n +2) N .If there hold (1) - (7) then:(8) The equiaffine mean curvature H of Σ r ( F, Ω) satisfies n H =( n + 2) sgn( U ( F )) | U ( F ) | − ( n +1) / ( n +2) ( H ( F ) − N p µ p ) . (2.43) (9) If Ω is simply connected, there is G ∈ C ∞ (Ω) satisfying dG = − ρ . Then G is locallyconstant on each level set Σ r ( F, Ω) and satisfies U ( G ) = sgn U ( F ) , and so | U ( G ) | = 1 , in Ω .Proof. Skew-symmetrizing (2.40) shows dρ = µ ∧ ρ . Consequently L W ρ = ι ( W ) dρ = µ ( W ) ρ − µ ,which implies the equivalence of (2) and (5), and there holds ρ ∧ L W ρ = µ ∧ ρ = dρ. (2.44)Since the flow of a vector field X preserves the codimension one smooth foliation annihilated bythe one-form θ if and only if θ ∧ L X θ = 0, the equivalence of (1)-(4) is immediate from (2.44).By (2.33), W ∧ N = 0 if and only if Z = 0. This shows the equivalence of (6) and (7). Since Z p Π ip = U ( F ) µ i − N p µ p F i , it also shows that W ∧ N = 0 if and only if µ ∧ ρ = 0. This shows theequivalence of (3) and (6). If there hold (1)-(7), then k ip µ p = 0, and (2.43) follows from (2.35).If Ω is simply connected, then there exists G ∈ C ∞ (Ω) such that G i = − ρ i = | U ( F ) | − / ( n +2) F i .Hence G ij = | U ( F ) | − / ( n +2) ( F ij − F i µ j ) = | U ( F ) | − / ( n +2) ( F ij − U ( F ) − N p µ p F i F j ) , (2.45)the second equality by (3). Calculating U ( G ) using (2.5) and (2.45) and simplifying the result usingelementary determinantal operations yields U ( G ) = sgn U ( F ). (cid:3) Corollary 2.12.
Let F be a C ∞ function defined on an open subset of R n +1 and suppose Ω is aconnected open set on which U ( F ) is equal to a nonzero constant κ . Then the equiaffine normal ofthe level set Σ r ( F, Ω) is W i = sgn( κ ) | κ | − ( n +1) / ( n +2) N i and its equiaffine mean curvature H satisfies n H = ( n + 2) sgn( κ ) | κ | − ( n +1) / ( n +2) H ( F ) . (2.46) Suppose there is an open domain ∆ ⊂ R n and a smooth map φ : ∆ → Ω such that φ is a dif-feomorphism onto its image and F ◦ φ ( x ) = t for all x ∈ ∆ for some t ∈ F (Ω) . Define ϕ : R × ∆ → Ω by ϕ ( t, x ) = Φ( −| κ | − / ( n +2) t, φ ( x )) where Φ( · , · ) is the maximal flow of W satisfying ddt Φ( t, p ) = W Φ( t,p ) . Then ddt ϕ ( t, x ) = W ϕ ( t,x ) , so ϕ solves the affine normal flow and F ◦ ϕ ( t, x ) = t + t , so that ϕ ( t, · ) is a parameterization of a subset of Σ t + t ( F, Ω) by an opensubset of ∆ .Proof. All the claims except for the final one regarding the affine normal flow follow from specializingLemma 2.11. By definition ddt Φ( t, p ) = W Φ( t,p ) , so ddt F ◦ ϕ ( t, x ) = dF ( W Φ( −| κ | − / ( n +2) ,φ ( x )) ) = 1 forall ( t, x ) in the domain of definition of Φ. Hence F ◦ ϕ ( t, x ) = t + t . (cid:3) Corollary 2.12 gives a criterion that is used in Section 3 to check that the examples constructedthere have equiaffine mean curvature zero; by Corollary 2.12, if U ( F ) is constant on Ω and H ( F )is constant on each level set Σ r ( F, Ω), then each level set Σ r ( F, Ω) has constant equiaffine meancurvature. In particular, if, on Ω, H ( F ) vanishes and U ( F ) equals a nonzero constant, then eachlevel set Σ r ( F, Ω) has equiaffine mean curvature zero.Lemma 2.11 shows that if U ( F ) is nonvanishing and locally constant on the level sets of F , thena level set of F is carried into another level set by the affine normal flow equal initially to the affinenormal along the first level set. Combining (9) of Lemma 2.11 and Corollary 2.12 shows that inthis case there is a second function G , a primitive of a constant multiple of − ρ , locally constant onthe level sets of F , and so that G ( x ) − G ( y ) is the the time the affine normal flow takes to movethe level set of F containing y to that containing x . All these statements are local, but in concretesituations they can make sense globally. Remark 2.10.
A nondegenerate hypersurface is an affine sphere if its equiaffine shape operator isa multiple of the identity. By the Gauss-Codazzi relations this multiple is necessarily a constant,and the affine sphere is proper or improper as it is nonzero or zero. QUIAFFINE GEOMETRY OF LEVEL SETS 15
By a theorem of S. Y. Cheng and S. T. Yau, on a proper open convex cone Ω ⊂ R n +1 there is aunique C ∞ function F solving H ( F ) = e F , tending to ∞ at the boundary of Ω, and such that F ij isa complete Riemannian metric on Ω. This F is − ( n +1)-logarithmically homogeneous and so satisfies U ( F ) = ( n + 1) H ( F ) = ( n + 1) e F . See [8] for details. Consequently, F and Ω satisfy the conditionsof Lemma 2.11. The function G of (9) of Lemma 2.11 is G = − − ( n + 2)( n + 1) − / ( n +2) e − F/ ( n +2) .It satisfies U ( G ) = 1 and H = n +2 n H ( G ) = − ( n + 1) − ( n +1) / ( n +2) e F/ ( n +2) = n +22( n +1) G − , (2.47)so its level sets have constant negative equiaffine mean curvature and are translated by the affinenormal flow. In fact the level sets of G (or F ) are affine spheres, although this does not followfrom Lemma 2.11. The examples constructed in Section 3 show that the level sets of a function G such that U ( G ) is constant and H ( G ) is constant along the levels of G can have equiaffine meancurvature zero without being improper affine spheres. However, for these examples H ( G ) vanishes,and I do not know any example of such a G for which H ( G ) is nonvanishing and the level sets of G are not affine spheres. It would be interesting to know if such examples exist. More precisely: if thelevel sets of a function F satisfying | U ( F ) | = 1 are strictly convex, must they be affine spheres? ♦ Example 2.11 illustrates some aspects of the notational conventions used here, as well as somesubtleties related to Lemma 2.5, in particular to connected components of level sets and the sig-natures of their equiaffine metrics. As is explained in [9], the polynomial considered in Example2.11 is best understood in the general context of prehomogeneous vector spaces, as the fundamentalrelative invariant of a real form of a prehomogeneous group action.
Example 2.11.
Using the Euclidean metric δ ij on R n , identify the space S ( R n ) of symmetricbilinear forms on R n with the space S n of symmetric n × n matrices. Endow S n with the Riemannianmetric (also denoted δ ) δ ( X, Y ) = tr XY having Levi-Civita connection ∇ and induced volume formΨ. For X ∈ S n , define coordinates x ij , 1 ≤ i ≤ j ≤ n , by X = P ni =1 x ii e ii + P ≤ i 1. By 1-3 above,the remaining eigenvalues of Hess P E p are 1, with multiplicity 2 p ( n − p ), and − 1, with multiplicity (cid:0) n − p (cid:1) + (cid:0) p (cid:1) . Hence the positive and negative inertia indices of (Hess P ) E p are 2 p ( n − p ) + 1 and (cid:0) n +12 (cid:1) − − p ( n − p ). Additionally, it follows that H ( P )( E p ) = ( − n +12 ) − ( n − 1) = ( − n ( n − − P ( E p )) ( n +1)( n − / . (2.50)The connected component of the identity, G , of the general linear group of R n acts on S n linearly byconjugation. Write ρ : G → GL ( S n ) for the induced linear representation. It is straightforward tocheck that the linear transformation ρ ( g ) of S n induced by g ∈ G satisfies det ρ ( g ) = (det g ) n +1 (itsuffices to check this for g a multiple of the identity). From (2.7) of Lemma 2.1 and ( ρ ( g ) · P )( X ) = P ( ρ ( g − ) X ) = (det g ) − P ( X ) it follows that ρ ( g ) · H ( P ) = (det ρ ( g )) H ( ρ ( g ) · P )= (det g ) n +1) H ((det g ) − P ) = (det g ) (2 − n )( n +1) H ( P ) . (2.51)By (2.51), H ( P ) and P ( n +1)( n − / are homogeneous polynomials of the same degree transformingidentically under the action of G on S n . As this action has an open orbit containing E p , with (2.50)this implies H ( P ) = ( − n ( n − − P ) ( n +1)( n − / .If F has positive homogeneity λ on R n , then ( λ − N i = ( λ − U ij F j = U ij F jk X k = H ( F ) X i and ( λ − U ( F ) = ( λ − N j F j = λ H ( F ) F , where X is the Euler vector field on R n . Because P has positive homogeneity n on S n , there results ( n − U ( P ) = nP H ( P ) and the transversal N equals ( n − − H ( P ) X . Hence U ( P ) and H ( P ) are nonzero and locally constant on the level set { X ∈ S n : P ( X ) = 1 } , and so it follows from (2.20) that along this level set the equiaffine normalis a negative constant times X . From (1) of Lemma 2.5 and Lemma 2.6 it follows that on thelevel set of P contained in the connected component C p the positive and negative inertial indicesof the equiaffine metric are 2 p ( n − p ) and (cid:0) n +12 (cid:1) − − p ( n − p ). Note that when p = 0 thisyields a negative definite metric and, by (2.43), the level set has positive equiaffine mean curvature.This reflects that, as can be checked using (2.48), log P is concave on C . Replacing P on C bythe function − P reverses the signature of the induced metric and the sign of the equiaffine meancurvature. Since U ( P ) is a multiple of P n − , it vanishes exactly on the locus of degenerate matricesin S n , and a connected component of the subset of S n where U ( P ) does not vanish equals one ofthe subspaces comprising matrices of a fixed nondegenerate signature. QUIAFFINE GEOMETRY OF LEVEL SETS 17 Because the equiaffine normal is a multiple of X , the connected components of the nonzerolevel sets of P are affine spheres. From (2.25) it follows that H ( − log P ) = P − n − . As a con-sequence, the function F of Remark 2.10, that solves H ( F ) = e F on the convex cone C , is F = − n +12 (cid:0) log P − n log n +12 (cid:1) . ♦ Another definition of the equiaffine normal. Let Σ be a co-orientable nondegenerateimmersed hypersurface in R n +1 . A transversal N along Σ determines on Σ the induced volume form ι ( N )Ψ given by interior multiplication with the volume form Ψ on R n +1 , and the volume density vol h determined by h and the orientation consistent with ι ( N )Ψ. By definition vol h = | det h | . Nextthere is given a second definition of a vector field W transverse to Σ that is manifestly equiaffinelycovariant, and it is shown that the transversal so defined coincides with the equiaffine normal asdefined in the textbook [14], where it is defined up to co-orientation by the requirements that thevolume densities | ι ( W )Ψ | and vol h be the same and be parallel with respect to the connection ∇ induced by b ∇ via W . This characterization of the equiaffine normal is used later to prove Theorem2.15, that shows that the transversal (2.33) of Theorem 2.8 is the equiaffine normal.The approach described now has as a side benefit a somewhat more general construction, namelyit attaches to nondegenerate immersed hypersurface in an ( n +1)-dimensional manifold M equippedwith an affine connection b ∇ a distinguished transverse line field that transforms covariantly underthe action of the group of affine automorphisms of ( M, b ∇ ). This was described previously in section4 of [7], but is hidden there in a still more general context requiring substantially more preparation,so it is recapitulated here.It is straightforward to check that, on an n -manifold, given a pair ([ ∇ ] , [ h ]) comprising a projec-tive structure [ ∇ ] and a conformal class [ h ] of pseudo-Riemannian metrics there is a unique repre-sentative ∇ ∈ [ ∇ ] that satisfies nh pq ∇ p h qi = h pq ∇ i h pq for every h ∈ [ h ]. This ∇ is said to be aligned with respect to [ h ]. The associated density-valued tensor H ij = | det h | − /n h ij is conformally invari-ant in the sense that it does not depend on the choice of h ∈ [ h ]. Alternatively, the aligned represen-tative is characterized by the equivalent requirement that there vanish all contractions of ∇ i H jk withthe dual density-valued bivector H ij = | det h | /n h ij . This makes the independence of the choice of h obvious. If ˜ ∇ = ∇ +2 γ ( i δ j ) k , then h pq ( n ˜ ∇ p h qi − ˜ ∇ i h pq ) − h pq ( n ∇ p h qi −∇ i h pq ) = ( n +2)(1 − n ) γ i .The alignment condition thus determines γ i , and so determines a unique representative of [ ∇ ].The description of the affine normal to be given now can be summarized as follows. A co-orientednondegenerate immersed hypersurface Σ in a manifold M equipped with an affine connection b ∇ carries the conformal structure induced by the second fundamental form and the co-orientationand the class of affine connections induced via all possible choices of transverse vector fields. Theconnections induced on Σ by b ∇ via transverse vector fields spanning the same line field are thesame, so each transverse line field induces on Σ a connection ∇ . A distinguished transverse linefield is determined by the requirement that the induced connection ∇ be aligned with respect to[ h ]. When there is an b ∇ -parallalel volume form Ψ on M , a distinguished transverse vector field W is determined by the requirement | ι ( N )Ψ | = | vol h | . If b ∇ is moreover projectively flat, then ∇ preserves the volume density of the representative h ∈ [ h ] corresponding to W . Lemma 2.13. Let Σ be a nondegenerate immersed hypersurface in an ( n +1) -dimensional manifold M equipped with a projectively flat affine connection b ∇ . Let N be a vector field defined in aneighborhood B in M of some point of Σ and transverse to Σ along B ∩ Σ and let ∇ , h , and τ be theconnection, metric, and connection one-form associated with b ∇ and N . Then ∇ [ I h J ] K = − τ [ I h J ] K .Proof. Because b ∇ is projectively flat, for vector fields X , Y , and Z tangent to Σ, the curvatureˆ R ( X, Y ) Z = ([ b ∇ X , b ∇ Y ] − b ∇ [ X,Y ] ) Z is tangent to Σ. On the other hand, evaluating ˆ R ( X, Y ) Z using(2.1) and projecting the result along N shows that ∇ [ I h J ] K = − τ [ I h J ] K . (cid:3) Theorem 2.14. Let Σ be a nondegenerate immersed hypersurface in an ( n + 1) -dimensional man-ifold M equipped with an affine connection b ∇ .(1) There is a unique line field W transverse to Σ such that the connection ∇ induced on Σ by b ∇ and W is aligned with respect to the metric k induced on an open U ⊂ Σ by anynonvanishing section of W over U .(2) If Σ is co-oriented and Ψ is a volume form on M preserved by b ∇ then there is a uniqueco-oriented section W of W such that the volume density | ι ( W )Ψ | equals the volume density | vol k | of k .(3) If Σ is co-oriented and b ∇ is projectively flat and preserves the volume form Ψ , then theconnection ∇ induced on Σ by b ∇ via the transversal W of (2) preserves the induced volumedensities | ι ( W )Ψ | and | vol k | .(4) If M = R n +1 and b ∇ and Ψ are the standard flat affine connection and standard volumeform then the transversal W of (2) is the equiaffine normal. The line field W of (1) is the affine normal distribution of Σ (with respect to b ∇ ). Note thatits definition does not suppose that Σ is co-orientable. The transverse vector field W of (2) is the equiaffine normal vector field of Σ (with respect to b ∇ and Ψ).The Levi-Civita connection of a pseudo-Riemannian metric of constant nonzero curvature isprojectively flat but not flat, and so by (1)-(3) it makes sense to speak of the affine normal distri-bution and equiaffine normal of a nondegenerate immersed hypersurface in manifold equipped witha constant curvature metric. Proof. Let U ⊂ Σ be an open neighborhood on which there is a smooth vector field N transverseto Σ. Any other transversal to Σ on U has the form ˜ N = a ( N + Z ) for a smooth function a notvanishing on U and a vector field Z tangent to U . The second fundamental form ˜ h , connection ˜ ∇ ,and connection one-form ˜ τ determined by ˜ N and b ∇ are related to h , ∇ , and τ by˜ h IJ = a − h IJ , ˜ ∇ = ∇ − h IJ Z K , ˜ τ I = τ I + a − da I + h IP Z P . (2.52)It follows from (2.52) that˜ h P Q ˜ ∇ I ˜ h P Q = h P Q ∇ I h P Q + 2 Z P h IP − na − da I , ˜ h P Q ˜ ∇ P ˜ h QI = h P Q ∇ P h QI + ( n + 1) Z P h IP − a − da I , (2.53)where h IJ and ˜ h IJ are the symmetric bivectors inverse to h IJ and ˜ h IJ , respectively. By (2.52) and(2.53), for any constants α , β , and γ , there holds α ˜ τ I + β ˜ h P Q ˜ ∇ P ˜ h QI + γ ˜ h P Q ˜ ∇ I ˜ h P Q = ατ I + βh P Q ∇ P h QI + γh P Q ∇ I h P Q + ( α − β − nγ ) a − da I + ( α + ( n + 1) β + 2 γ ) Z P h IP (2.54)If α = β + nγ , so that (2.54) does not depend on a , then β (˜ τ I + ˜ h P Q ˜ ∇ P ˜ h QI ) + γ ( n ˜ τ I + ˜ h P Q ˜ ∇ I ˜ h P Q )= β ( τ I + h P Q ∇ P h QI ) + γ ( nτ I + h P Q ∇ I h P Q ) + ( n + 2)( β + γ ) Z P h IP . (2.55)Since (2.55) does not depend on a , as long as β = − γ , the direction of a transversal on U isdetermined by requiring that the left side of (2.55) vanish. The condition β = − γ is explainedas follows. The proof of Lemma 2.13 applied to the not necessarily projectively flat connection b ∇ shows that ∇ [ I h J ] K + τ [ I h J ] K is expressible in terms of the curvature of b ∇ , and tracing this relationshows that ( nτ I + h P Q ∇ I h P Q ) − ( τ I + h P Q ∇ P h QI ) is determined by the curvature of b ∇ . Hencethere is no freedom to choose the value of this quantity. When β = − γ , the transversal determinedby the vanishing of the left side of (2.55) depends only on the image of ( β, γ ) in the projective line, QUIAFFINE GEOMETRY OF LEVEL SETS 19 so there is one parameter of freedom in the choice of a transverse line field. Among these possiblenormalizations there is one that is distinguished in that the compatibility condition determiningit involves only ∇ and h and does not involve τ ; this is the condition given by β = − nγ . Thiscorresponds to the identity n ˜ h P Q ˜ ∇ P h QI − ˜ h P Q ˜ ∇ I h P Q = nh P Q ∇ P h QI − h P Q ∇ I h P Q + ( n + 2)( n − Z P h IP , (2.56)and requiring that the left side of (2.56) vanish is exactly requiring that ˜ ∇ be aligned with respectto [ h ]. This yields Z P h P I = n +2)(1 − n ) (cid:0) nh P Q ∇ P h QI − h P Q ∇ I h P Q (cid:1) . (2.57)The span W of ˜ N is well defined, independently of the remaining freedom, which is the choiceof a . Since around any point of Σ there can be found an open subset U ⊂ Σ on which there isa transversal, and since by the uniqueness just proved the line fields constructed on overlappingneighborhoods agree on the overlaps, the transverse line field W is defined globally on Σ, evenin the case that Σ is not co-orientable. If Σ is co-orientable then the transversal ˜ N = a ( N + Z )spanning W can be taken to be globally defined. Since det ˜ h = a − n det h and ι ( ˜ N )Ψ = aι ( N )Ψ, | vol ˜ h /ι ( ˜ N )Ψ | = | a | − ( n +2) / | vol h | / | ι ( N )Ψ | , so requiring that the induced volume density | ι ( ˜ N )Ψ | equal the volume density | vol ˜ h | of the associated metric determines a up to sign, which is fixed bychoosing a so that W is co-oriented.If b ∇ is projectively flat, then, by Lemma 2.13, nτ I + h P Q ∇ I h P Q = τ I + h P Q ∇ P h QI , (2.58)and similarly for ˜ ∇ , ˜ h , and ˜ τ . With (2.58), the vanishing of the left side of (2.56) is equivalent to n ˜ τ I + ˜ h P Q ˜ ∇ I ˜ h P Q = 0 = ˜ τ I + ˜ h P Q ˜ ∇ P ˜ h QI , (2.59)while (2.57) becomes Z P h P I = − n +2 (cid:0) nτ I + h P Q ∇ I h P Q (cid:1) . (2.60)Now suppose b ∇ is projectively flat and preserves a volume form Ψ. Let k be the metric and let ∇ and τ be the connection and connection one-form associated with the normal W determined by(2). By (2.2) and (2.60), − nτ I | vol k | = k AB ∇ I k AB | vol k | = ∇ I | vol k | = ∇ I | ι ( W )Ψ | = τ I | ι ( W )Ψ | = τ I | vol k | . (2.61)Hence 0 = − nτ I | vol k | = ∇ I | vol k | , so that τ vanishes and | vol k | is ∇ -parallel. Now suppose M = R n +1 and ( b ∇ , Ψ) is the standard flat equiaffine structure. The usual definition of the equiaffinenormal given in [14] is that the volume densities | ι ( W )Ψ | and | vol k | coincide and are preserved bythe connection ∇ induced on Σ by b ∇ via W , so (4) is immediate from (2). (cid:3) Theorem 2.15 shows that the equiaffinely covariant transversal of Theorem 2.8 coincides withthe equiaffine normal. Theorem 2.15. Let F be a C ∞ function defined on an open subset of R n +1 . Let Ω be a con-nected component with nonempty interior of the region on which U ( F ) does not vanish. Theequiaffine normal of a nonempty level set Σ r ( F, Ω) consistent with the co-orientation determinedby − sgn( U ( F )) N i is given by the expression (2.20) .Proof. Let h , ∇ , and τ be respectively the second fundamental form, connection, and connectionone-form determined according to (2.1) by the transversal N i = U ip F p and b ∇ . The most generalvector field transverse to the Σ r ( F, Ω) has the form W = a ( N + Z ) for a nonvanishing smoothfunction a and a vector field Z tangent to the Σ r ( F, Ω). The restriction to Σ r ( F, Ω) of the tensor h ij = − U ( F ) − Π ij of (2.12) is the second fundamental form determined by N i . By (2.37) thereholds F p b ∇ i N p = d U ( F ) i − H ( F ) F i , so that F p ( b ∇ i N p − (( n + 2) µ i − U ( F ) − H ( F ) F i ) N j ) = 0and therefore the connection one-form determined by N i is the restriction of the one-form τ i =( n + 2) µ i − U ( F ) − H ( F ) F i . This can be written as the identity τ I = ( n + 2) µ I , that refers only tothe tangential directions. By (5) of Lemma 2.5, the volume density | vol h | induced on Σ r ( F, Ω) by h satisfies | vol h | = | U ( F ) | − ( n +1) / | ι ( N )Ψ | and so, by definition of µ and (2.2), h AB ∇ I h AB = 2 | vol h | − ∇ I | vol h | = − ( n + 1)( n + 2) µ I + 2 τ I . (2.62)Substituting (2.62) in (2.60) and using τ I = ( n + 2) µ I yields − ( n + 2) Z A h IA = nτ I + h AB ∇ I h AB =( n + 2) µ I , so Z P h P I = − µ I . This is the condition defining the vector field also called Z i in(2.33), so the two are the same. By the proof of Theorem 2.14, | a | = | vol h /ι ( N )Ψ | / ( n +2) = | U ( F ) | − ( n +1) / ( n +2) . It follows that the equiaffine normal consistent with the co-orientation deter-mined by − sgn( U ( F )) N i is W i = − sgn( U ( F )) | a | ( N i + Z i ), which is (2.33). (cid:3) Equiaffine mean curvature zero, n -dimensional hypersurfaces ruled by n -dimensional planes This section describes a general construction of hypersurfaces that generalize the usual helicoids.These are 2 n -dimensional hypersurfaces ruled by n -planes and having equiaffine mean curvaturezero. This construction directly generalizes one for surfaces given by A. Mart´ınez and F. Milanin [12], but the author found it by considering modifications of the homogeneous polynomialsconstructed by Gordan-N¨other in [10] (these were mentioned in Example 2.6) that while havingvanishing Hessian determinant have linearly independent partial derivatives.At places in this section it is convenient to use slightly abusive coordinate dependent notation.For clarity, the translation to the abstract index notation used throughout the paper is indicatedwhere confusion could arise.3.1. General construction. Regard R n +1 as equipped with the standard flat affine connection b ∇ and the parallel volume form Ψ given by the determinant. The pairing between R n +1 and thedual vector space R n +1 ∗ is written h · , · i : R n +1 × R n +1 ∗ → R . The same notation is used for thepairing induced on tensor powers. The space R n +1 ∗ is endowed with the dual connection inducedby b ∇ , also denoted by b ∇ , and the parallel ( n + 1)-form Ψ ∗ dual to Ψ (meaning that h Ψ , Ψ ∗ i = 1).Let M be an open smooth connected submanifold of R n , so M is the nonempty interior of adomain in R n , or the entirety of R n . Let A : M → R n +1 be a smooth map and fix Q ∈ C ∞ ( M ).Define a smooth function F : M × R n +1 ∗ → R by F ( u, x ) = h A ( u ) , x i + Q ( u )(3.1)for u ∈ M and x ∈ R n +1 ∗ . Equip M with the flat affine connection D and parallel volume form ω induced by the standard flat connection and volume form on R n . Equip M × R n +1 ∗ with the (flat)product connection ∇ determined by D and b ∇ , and the volume form written, with a slight abuseof notation, as Υ = ω ∧ Ψ ∗ . The goal of this section is to obtain conditions on the map A thatguarantee that the level sets of F are nondegenerate hypersurfaces with equiaffine mean curvaturezero with respect to the equiaffine structure ( ∇ , Υ).Recall the definition of equiaffine coordinates from the beginning of section (2.1) and fix coor-dinates x , . . . , x n +1 on R n +1 that are equiaffine with respect to b ∇ and Ψ. Let x , . . . , x n +1 becoordinates on R n +1 ∗ that are equiaffine with respect to b ∇ and Ψ ∗ and dual to the equiaffinecoordinates x , . . . , x n +1 . This means that the one-forms dx , . . . , dx n +1 on R n +1 ∗ constitute acoframe dual to that comprising dx , . . . , dx n +1 . Because of this duality, a parallel one-form on R n +1 ∗ can be regarded as a parallel vector field on R n +1 , and so it is convenient to write ∂ x i for dx i . Let u , . . . , u n be equiaffine coordinates on ( M, D, ω ). While the coordinates u , . . . , u n are QUIAFFINE GEOMETRY OF LEVEL SETS 21 determined up to automorphisms of ( D, ω ), in what follows it will be more convenient to think of D and ω as determined by the choice of coordinates. Let a ( u , . . . , u n ) , . . . , a n +1 ( u , . . . , u n ) bethe components of A with respect to the parallel frame ∂ x , . . . , ∂ x n +1 . In these coordinates F isgiven by F ( u, x ) = P n +1 i =1 x i a i ( u ) + Q ( u ).The Hessian of F is defined with respect to the product connection ∇ and the volume form Ω.The differential and Hessian of F are dF u,x = h dA u , x i + dQ u + h A ( u ) , dx i = n +1 X i =1 x i da i + dQ + n +1 X i =1 a i dx i , Hess F u,x = h DdA u , x i + DdQ u + 2 h dA u , dx i = n +1 X i =1 ( da i ⊗ dx i + dx i ⊗ da i ) + n +1 X i =1 x i Dda i + DdQ. (3.2)Let a iI = ∂a i ∂u I = da i ( ∂ u I ) and write da i = P nI =1 a iI du I and A I = dA ( ∂ u I ). With respect to thechosen equiaffine coordinate frames, dA can be regarded as an ( n + 1) × n matrix, dA = da ... da n +1 = a . . . a n ... . . . ... a n +11 . . . a n +1 n = (cid:0) A . . . A n (cid:1) (3.3)and the Hessian of F as a matrix with block formHess F = (cid:18)P n +1 i =1 x i Dda i + DdQ dA t dA (cid:19) , (3.4)where the superscript t indicates the matrix transpose.Define a vector field V on M × R n +1 ∗ by V = n +1 X i =1 ( − i +1 dA ( i ) ∂ x i (3.5)where dA ( i ) is the determinant of the n × n matrix obtained from dA as in (3.3) by deleting the i throw. A vector field on R n +1 ∗ can be regarded as a one-form on R n +1 and, via this identification,for fixed u ∈ M , V is identified with the one-form which when paired with X ∈ R n +1 yields h X, V i = Ψ( X, A , . . . , A n ) = (cid:12)(cid:12) X A . . . A n (cid:12)(cid:12) = (cid:12)(cid:12) X dA (cid:12)(cid:12) , (3.6)where vertical bars indicate the determinant of a matrix, the determinant is defined relative to thevolume form Ψ, and the various notations in (3.6) are synonymous.Let X be the radial (position) vector field on R n +1 generating dilations centered at the originand define the n -form µ = ι ( X )Ψ. If X , . . . , X n are vector fields on R n +1 , then µ ( X , . . . , X n ) = Ψ( X , X , . . . , X n ) = (cid:12)(cid:12) X X . . . X n (cid:12)(cid:12) , (3.7)where vertical bars indicate the determinant of a matrix. An immersed codimension one submani-fold of R n +1 is centroaffine if it does not contain the origin and is everywhere transverse to X . Animmersion A : M → R n +1 of an n -manifold is centroaffine if A ( M ) is a centroaffine submanifold.Equivalently, the pullback A ∗ ( µ ) is a volume form on M . If φ : M ′ → M is a diffeomorphism, then φ ∗ A ∗ ( µ ) is nonvanishing if and only if A ∗ ( µ ) is nonvanishing, so the immersion A is centroaffine ifand only if the immersion A ◦ φ : M ′ → R n +1 is centroaffine.Define a one-form β on M × R n +1 ∗ by β = h A, dx i = n +1 X i =1 a i dx i , (3.8) and let Ω = dβ . Straightforward computations using da ∧ · · · ∧ da n +1 = 0 show β ∧ Ω n = ( − n ( n +1) / n ! A ∗ ( µ ) ∧ Ψ ∗ = ( − n ( n +1) / n ! (cid:12)(cid:12) A dA (cid:12)(cid:12) Υ , (3.9)and β ( V ) = (cid:12)(cid:12) A dA (cid:12)(cid:12) . Since Ω( V, ∂ u I ) = (cid:12)(cid:12) A I dA (cid:12)(cid:12) = 0, there holds ι ( V )Ω = 0. Hence (cid:12)(cid:12) A dA (cid:12)(cid:12) Ω n = ι ( V )( β ∧ Ω n ) = ( − n ( n +1) / n ! (cid:12)(cid:12) A dA (cid:12)(cid:12) ι ( V )Υ . (3.10) Lemma 3.1. Let M be an open smooth connected submanifold of R n equipped with the induced flataffine connection D and the induced parallel volume form ω . Given a smooth map A : M → R n +1 and Q ∈ C ∞ ( M ) define the smooth function F : M × R n +1 ∗ → R by F ( u, x ) = h A ( u ) , x i + Q ( u ) for u ∈ M and x ∈ R n +1 ∗ . Equip R n +1 ∗ with the standard flat affine connection b ∇ and parallel volumeform Ψ ∗ , and define the Hessian Hess F with respect to the product flat connection ∇ determinedby D and b ∇ and the volume form Υ = ω ∧ Ψ ∗ . Define a vector field V on M × R n +1 ∗ by (3.5) .Then:(1) V is contained in the radical of Hess F .(2) The adjugate tensor adj Hess F equals ( − n V ⊗ V and the vector field N i = U ij F j equals N = ( − n (cid:12)(cid:12) A dA (cid:12)(cid:12) V .(3) A is an immersion if and only if the rank of Hess F is n . In this case the vector field V generates the radical of Hess F .(4) The following are equivalent(a) A is a centroaffine immersion.(b) U ( F ) is nowhere vanishing on M × R n +1 ∗ .(c) The one-form β defined in (3.8) is a contact one-form.In this case, a nontrivial level set Σ of F is a smoothly immersed nondegenerate submanifoldof M × R n +1 ∗ co-oriented by N = ( − n (cid:12)(cid:12) A dA (cid:12)(cid:12) V , and adj Hess F = ( − n (cid:12)(cid:12) A dA (cid:12)(cid:12) − N ⊗ N . Moreover, the restriction to Σ of Ω = dβ is a symplectic form.(5) If there is a nonzero constant κ such that A ∗ ( µ ) = κω (3.11) (equivalently, β ∧ Ω n = ( − n ( n +1) / n ! κ Υ ) then A is a centroaffine immersion and, for anontrivial level set Σ = { ( u, x ) ∈ M × R n +1 ∗ : F ( u, x ) = t } of F , there hold:(a) Σ has zero equiaffine mean curvature and equiaffine normal W = ( − n +1 | κ | − n +1) / ( n +2) N = − sgn( κ ) | κ | − n/ ( n +2) V. (3.12) (b) The equiaffine shape operator S of Σ is nilpotent with square equal to zero, and ker S contains the tangent distribution T of a ruling of Σ by n -dimensional affine planesLagrangian with respect to the symplectic form Ω .(c) The ruling T of Σ is not cylindrical.(d) The equiaffine metric h of Σ has split signature.(e) The Reeb field of β is κ − V .(f ) Define Φ : R × R n × R n → M × R n +1 ∗ by Φ( t, r, s ) = ( r , . . . , r n , c ( t, r, s ) , . . . , c n +1 ( t, r, s )) , (3.13) where c i ( t, r, s ) = ( − i +1 κ − ( t − Q ( r )) dA ( i ) ( r ) + e i ( t, r, s ) , (3.14) for e ( t, r, s ) = s a ( r ) , e n +1 ( t, r, s ) = − s n a n ( r ) , and e i ( t, r, s ) = − s i − a i − ( r ) + s i a i +1 ( r ) for ≤ i ≤ n . Then F (Φ( t, r, s )) = t and for each t ∈ R for which Σ = { ( u, x ) ∈ M × R n +1 ∗ : F ( u, x ) = t } is nonempty, Φ( t, · , · ) : R n × R n → M × R n +1 , ∗ is a parameterization of Σ . QUIAFFINE GEOMETRY OF LEVEL SETS 23 (g) The map ϕ : R × R n × R n → M × R n +1 , ∗ defined by ϕ ( t, r, s ) = Φ( −| κ | / ( n +2) t, r, s ) solves the affine normal flow ddt ϕ ( t, r, s ) = W ϕ ( t,r,s ) . (3.15) Proof. Since dF ( ∂ x i ) = a i , dF = 0 if and only if A is the zero map. Hence if A is not the zero map,every level set of F is regular, so smoothly immersed.Since Hess F ( V, · ) = P n +1 i =1 ( − i +1 dA ( i ) da i , Hess F ( V, X ) vanishes for any X tangent to R n +1 ∗ .On the other hand,Hess F ( V, ∂ u I ) = n +1 X i =1 ( − i +1 dA ( i ) a iI = (cid:12)(cid:12) A I A . . . A n (cid:12)(cid:12) = 0 . (3.16)Hence V is contained in the radical of Hess F , meaning Hess F ( V, · ) = 0, and Hess F is degeneratewith rank no greater than 2 n .To analyze the rank and adjugate of Hess F it is most straightforward to examine the matrixrepresentation (3.4). Consider a 2 n × n submatrix of (3.4). If this submatrix is obtained bydeleting a row or column not intersecting the null block in the lower right corner, then it containsan ( n + 1) × ( n + 1) null block, and its determinant is 0. If this submatrix is obtained by deletinga row and column intersecting the null block in the lower right corner, then it contains four n × n blocks, the lower right of which is null, and its determinant is the product of the determinants of theantidiagonal blocks multiplied by ( − n = ( − n . These antidiagonal blocks are n × n submatricesof dA and its transpose and so the corresponding minor equals ( − n dA ( i ) dA ( j ) . Since the entryin the row n + j and column n + i of the matrix representing adj Hess F is this minor multiplied by( − ( n + i )+( n + j ) = ( − i + j , it equals ( − i + j + n dA ( i ) dA ( j ) . This shows thatadj Hess F = (cid:18) C (cid:19) = ( − n V ⊗ V (3.17)where the ( n + 1) × ( n + 1) matrix C has components C ij = ( − i + j + n dA ( i ) dA ( j ) . (In the abstractindex notation used in most of this paper, (3.17) would be written U ij = ( − n V i V j .)If u , . . . , u n are equiaffine coordinates on ( M, D, ω ), then A ∗ ( µ ) = (cid:12)(cid:12) A dA (cid:12)(cid:12) du ∧ · · · ∧ du n , or,equivalently, (cid:12)(cid:12) A dA (cid:12)(cid:12) = (cid:12)(cid:12) A A . . . A n (cid:12)(cid:12) = Ψ( A, A , . . . , A n ) = A ∗ ( µ )( ∂ u , . . . , ∂ u n ) , (3.18)so that, from (3.6), it follows that dF ( V ) = (cid:12)(cid:12) A dA (cid:12)(cid:12) . Consequently, multiplying (3.17) by dF yields N = ( − n dF ( V ) V = ( − n (cid:12)(cid:12) A dA (cid:12)(cid:12) V, (3.19) U ( F ) = dF ( N ) = ( − n dF ( V ) = ( − n (cid:12)(cid:12) A dA (cid:12)(cid:12) . (3.20)That A be an immersion means that dA has rank n everywhere. The preceding shows directlythat the rank of dA is n if and only if Hess F has a nonvanishing 2 n × n minor, so has rank2 n . Precisely, V is nowhere vanishing if and only if dA has rank n everywhere, and becauseadj Hess F = ( − n V ⊗ V this holds if and only if adj Hess F has rank 1 everywhere, which isequivalent to Hess F having rank 2 n everywhere.Consequently, if A is an immersion, then the radical of Hess F has dimension one, and sincethe radical contains the nowhere vanishing vector field V , it must be that V generates the radical,meaning that any vector X satisfying Hess F ( X, · ) = 0 is a multiple of V .By (3.20), U ( F ) is nowhere vanishing if and only if A ∗ ( µ ) is nowhere vanishing, so A is acentroaffine immersion if and only if U ( F ) is nowhere vanishing. By (3.9) this holds if and onlyif β is a contact one-form. In this case, by Lemma 2.5, there hold the claims in (4) regarding Σ and N . Together (3.19) and (3.17) show adj Hess F = ( − n (cid:12)(cid:12) A dA (cid:12)(cid:12) − N ⊗ N . By (3.10),Ω n = ( − n ( n +1) / n ! ι ( V )Υ, and so the restriction of Ω to Σ is nondegenerate. This shows (4).If there is a constant κ = 0 such that there holds (3.11), then, by (3.20), U ( F ) = κ is constant,so, by Corollary 2.12, the level sets of F have equiaffine mean curvature zero and equiaffine normalgiven by (3.12). It is apparent from the explicit form (3.5) of V that the equiaffine shape operator S is nilpotent. More precisely, its square is 0, for its image is contained in the span of the coordinatevector fields ∂ x , . . . , ∂ x n +1 , while its kernel contains these vector fields. Note that the equiaffinenormals are constrained to lie in a linear subspace.For each u ∈ M let c , . . . , c n be vectors in R n +1 spanning the kernel of the one-form A ( u ). Thecomponents c Ii of c I with respect to the equiaffine coordinates x i on R n +1 are functions of u alone.The linearly independent vector fields T I = P n +1 p =1 c Ip ∂ x p satisfy dF ( T I ) = 0, so are tangent to Σ.As ∇ T I T J = 0, the T I span a totally geodesic rank n distribution T tangent to Σ; its leaves are n -dimensional affine planes. Since W is a constant multiple of V and ∇ T I V = 0, T ⊂ ker S . Theannihilator Ann T of T is spanned by β and du , . . . , du n . Were T cylindrical, then Ann T wouldbe preserved by b ∇ . Since the du i are b ∇ -parallel, this can be only if there are smooth functions b I ( u ), 1 ≤ I ≤ n , such that b ∇ β = P nI =1 b I du I ⊗ β . Equivalently, ∂a i ∂u I = b I a i . Differentiating thisidentity yields ∂ a i ∂u I ∂u J = b I b J a i + a i ∂b I ∂u J . Hence a i ( ∂b I ∂u J − ∂b J ∂u I ) = 0. Were a i to vanish on an openset, then on this open set there would vanish (cid:12)(cid:12) A dA (cid:12)(cid:12) , a contradiction. If ¯ u is a point where nocomponent of A (¯ u ) vanishes, then the same holds in an open neighborhood U of ¯ u . Hence on U there holds ∂b I ∂u J = ∂b J ∂u I , so on some possibly smaller neighborhood of ¯ u , also to be called U , thereis a smooth function f such that b I = ∂f∂u I . Then ∂∂u I ( e − f a i ) = 0, so on U there are constants c i = 0 such that a i = c i e − f . However, this implies (cid:12)(cid:12) A dA (cid:12)(cid:12) = e − ( n +1) f (cid:12)(cid:12) c − b c . . . − b n c (cid:12)(cid:12) = 0,which is a contradiction. It follows that T is not cylindrical. For any vector field X tangent toΣ, ∇ X V is contained in the span of the vector fields ∂ x , . . . , ∂ x n +1 . Since V is also contained inthis span, and W is a constant multiple of V , S ( X ) is contained in the span of the vector fields ∂ x , . . . , ∂ x n +1 . Since then ∇ S ( X ) V = 0, the square of S is 0. Since ∇ T I T J = 0, the equiaffinemetric h satisfies h ( T I , T J ) = 0, so T is an n -dimensional h -isotropic subspace, which shows that h has split signature. Since Ω( T I , T J ) = 0, T is Lagrangian. Since β ( V ) = κ and ι ( V )Ω = 0, theReeb field of β is κ − V .That (3.13) satisfies F (Φ( t, r, s )) = t follows from the observation that P n +1 i =1 a i ( r ) c i ( t, r, s ) = t .To show (3.15), it is helpful to rewrite the parameterization (3.13) of Σ of (5f) in the following way.Let U I = ∂ u I and X i = ∂ X i . For 1 ≤ I ≤ n define Y I = a I +1 X I − a I X I +1 . ThenΦ( t, r, s ) = n X I =1 r I U I + n X I =1 s I Y I ( r ) + κ − ( t − Q ( r )) V = n X I =1 r I U I + n X I =1 s I Y I ( r ) + | κ | − / ( n +2) ( Q ( r ) − t ) W . (3.21)From (3.21) it is clear that ddt Φ( t, r, s ) = −| κ | − / ( n +2) W Φ( t,r,s ) , from which (3.15) follows. (cid:3) Remark 3.1. If A is an immersion and A ( u ) = 0 then the rank of Hess F is 2 n at ( u, x ) although U ( F ) vanishes at ( u, x ). This does not contradict Lemma 2.4 because at ( u, x ) the nontrivial vectorfield V is tangent to the level set of F through ( u, x ) and contained in the radical of Hess F , andso this level set is degenerate. Note also that such an immersion A is not centroaffine. ♦ QUIAFFINE GEOMETRY OF LEVEL SETS 25 Remark 3.2. The parameterization (3.13) of the level set Σ of (5f) or (3.21) of Lemma 3.1 can berewritten as Φ( t, r, s ) = (cid:18) C ( r ) (cid:19) s + (cid:18) rD ( t, r ) (cid:19) , (3.22)where the ( n + 1) × n matrix C ( r ) and the ( n + 1)-vector D ( t, r ) have components C iJ ( r ) = a i +1 ( r ) if i = J, − a i ( r ) if i = J + 1 , , D i ( t, r ) = ( − i +1 κ − ( t − Q ( r )) dA ( i ) ( r ) , (3.23)where 1 ≤ i ≤ n + 1 and 1 ≤ J ≤ n . From the representation (3.22) it is clear that Σ is ruled by n -planes. ♦ Lemma 3.2. Let M be an open smooth connected submanifold of R n equipped with the induced flataffine connection D and the induced parallel volume form ω . Equip R n +1 ∗ with the standard flataffine connection b ∇ and parallel volume form Ψ ∗ . Suppose Q ∈ C ∞ ( M ) and let A : M → R n +1 bea centroaffine immersion satisfying A ∗ ( µ ) = κω for a nonzero constant κ . A nonempty connectedcomponent Σ of a level set of the function F : M × R n +1 ∗ → R defined by F ( u, x ) = h A ( u ) , x i + Q ( u ) is an affine sphere if and only if the image of A is contained in a hyperplane.Proof. Since, by Lemma 3.1, the shape operator of Σ is nilpotent, if Σ is an affine sphere it isnecessarily improper. This means that the equiaffine normal W is parallel along the hypersurface.Since W is a constant multiple of the vector field V defined in (3.5), it follows that V is b ∇ -parallelalong Σ, and so the components ( − i +1 dA ( i ) of V are equal to constants p i , that can be viewedas the components of a constant vector p ∈ R n +1 ∗ . Then κ = (cid:12)(cid:12) A dA (cid:12)(cid:12) = h A ( u ) , p i , so the image ofthe centroaffine immersion A is contained in the hyperplane { v ∈ R n +1 : κ = h v, p i} .Now suppose A is a centroaffine immersion satisfying (cid:12)(cid:12) A dA (cid:12)(cid:12) = κ for a nonzero constant κ andhaving image contained in the hyperplane { v ∈ R n +1 : 1 = h v, p i} for a constant vector p ∈ R n +1 ∗ .First suppose that p = dx n +1 . Then a n +1 = 1, so dA = a . . . a n ... . . . ... a n . . . a nn . . . , κ = (cid:12)(cid:12) A dA (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a a . . . a n ... ... . . . ... a n a n . . . a nn . . . (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = ( − n dA ( n +1) . (3.24)Hence, for 1 ≤ i ≤ n , the minors dA ( i ) are zero, while dA ( n +1) is constant. This shows that V is aconstant vector, so parallel, so Σ is an improper affine sphere. For general p there is g ∈ Gl ( n + 1 , R )such that g ∗ p = dx n +1 , where g ∗ indicates the adjoint action defined by h u, g ∗ mu i = h g − u, µ i for v ∈ R n +1 and µ ∈ R n +1 ∗ . Then 1 = h A, p i = h gA, dx n +1 i . Since (cid:12)(cid:12) gA d ( gA ) (cid:12)(cid:12) = (det g ) (cid:12)(cid:12) A dA (cid:12)(cid:12) = κ det g , the preceding shows that d ( gA ) ( i ) = 0 for 1 ≤ i ≤ n and d ( gA ) ( n +1) = ( − n κ det( g ). Bythe Cauchy-Binet formula for minors, for 1 ≤ i ≤ n + 1, d ( gA ) ( i ) = P n +1 j =1 | g ( ij ) | dA ( j ) , where | g ( ij ) | is the determinant of the n × n submatrix of g obtained by deleting the i th row and j th column.This shows that the minors dA (1) , . . . , dA ( n +1) solve n + 1 constant coefficient linear equations, andso they must be constants. Hence V is parallel and Σ is an affine sphere. (cid:3) In order to obtain interesting examples from Lemma 3.1, it is necessary to solve the partialdifferential equation (3.11) for some constant κ = 0. Let φ : M → M ′ be a diffeomorphismsbetween open domains in R n . Then | dφ | (cid:12)(cid:12) A ◦ φ − d ( A ◦ φ − ) (cid:12)(cid:12) ◦ φ = (cid:12)(cid:12) A dA (cid:12)(cid:12) . (3.25) If φ is to be such that (cid:12)(cid:12) A ◦ φ − d ( A ◦ φ − ) (cid:12)(cid:12) = κ for some constant κ = 0, then it follows from(3.25) that it must be that | dφ | = κ − (cid:12)(cid:12) A dA (cid:12)(cid:12) . (3.26)Solving (3.26) is simply the problem of finding a diffeomorphism that pulls a given volume formback to a standard volume form. Beginning with J. Moser’s [13], solutions to several versions ofthis problem have been obtained, for various combinations of hypotheses regarding regularity, totalvolume, and the topology of the underlying spaces. Lemma 3.3. Let Σ ⊂ R n +1 be a centroaffine smoothly immersed codimension one submanifold forwhich there exists a smooth diffeomorphic parameterization A : M → Σ for some open connectedsmooth submanifold M of R n , let µ = ι ( X )Ψ , and let ω be the standard volume form on R n that ispreserved by the standard flat affine connection on R n . Then:(1) If the closure of M is a compact connected smooth manifold with boundary with respect tothe smooth structure induced from R n , then there is a diffeomorphism φ : M → M isotopicto the identity and extending to the identity on the boundary of the closure of M such that B = A ◦ φ : M → R n +1 satisfies B ∗ ( µ ) = κω .(2) If M has infinite volume with respect to ω , there exists a smooth embedding φ : R n → M such that B = A ◦ φ : M → R n +1 satisfies B ∗ ( µ ) = κω for some nonzero constant κ .Proof. In the setting of (1), by the main theorem of A. Banyaga’s [1] there exists a diffeomorphism φ : M → M isotopic to the identity and extending to the identity on the boundary of the closureof M such that φ ∗ A ∗ ( µ ) = vol A ∗ ( µ ) (Σ) ω .In the setting of (2), by a theorem of F. Schlenk (see Appendix B of [18]) there exists a smoothembedding φ : R n → M such that φ ∗ A ∗ ( µ ) = ω . (cid:3) A version of claim (1) with lower regularity assumptions could be obtained by using the morewell-known theorem of Dacorogna-Moser, [6], instead of the cited theorem of Banyaga.Lemma 3.3 means that if a centroaffine immersed submanifold of R n +1 admits a smooth param-eterization by an open connected submanifold of R n that either has infinite volume with respect tothe standard volume form ω on R n or has compact closure with infinitely smooth boundary, thenit admits a parameterization A by such a submanifold M such that A ∗ ( µ ) is a nonzero constantmultiple of the restriction to M of ω . Combining Lemmas 3.1 and 3.3 proves the following theorem. Theorem 3.4. Given an immersed hypersurface Σ in R n +1 everywhere transverse to the radialvector field X and diffeomorphic to a connected open submanifold M of R n that either has infinitevolume with respect to the standard volume form ω on R n or has compact closure with infinitelysmooth boundary, there exists a centroaffine parameterization A : M → Σ satisfying A ∗ ( µ ) = κω fora nonzero constant κ and the restriction of ω to M , and, for any Q ∈ C ∞ ( M ) , the level sets of thefunction F : M × R n +1 ∗ → R defined by F ( u, x ) = h A ( u ) , x i + Q ( u ) are equiaffine mean curvaturezero smooth submanifolds of M × R n +1 ruled by n -planes tangent to the kernel of S and for whichthe equiaffine shape operator S is nilpotent of order two. The one-form β = h A, dx i = P n +1 i =1 a i dx i is a contact one-form on M × R n +1 ∗ for which the Reeb field is a constant multiple of the equiaffinenormal of the level sets of F , and the restriction of dβ to each level set is a symplectic form withrespect to which the ruling of the level set is a Lagrangian foliation. Examples. When n = 1, the surfaces obtained from the construction of section 3.1 are asubset of those described in section two of [12].Let a ( t ) and b ( t ) be the components of an immersion A : I ⊂ R → R . Then (cid:12)(cid:12) A dA (cid:12)(cid:12) = ab ′ − ba ′ equals a nonzero constant κ if and only if a and b are linearly independent solutionsof a homogeneous linear second order differential equation f ′′ ( t ) + r ( t ) f ( t ) = 0. Suppose thatthere exist linearly independent twice differentiable solutions a ( t ) and b ( t ) defined for all t ∈ R . QUIAFFINE GEOMETRY OF LEVEL SETS 27 Then κ = ab ′ − a ′ b . Let Q ∈ C ∞ ( R ) and define F ( u, x, y ) = a ( u ) x + b ( u ) y + Q ( u ). Then dF = adx + bdy + ( a ′ ( u ) x + b ′ ( u ) y + Q ′ ( u )) du and b ∇ dF = ( a ′ ( u ) dx + b ′ ( u ) dy ) ⊗ du + du ⊗ ( a ′ ( u ) dx + b ′ ( u ) dy ) + ( xa ′′ ( u ) + yb ′′ ( u ) + Q ′′ ( u )) du ⊗ du . While H ( F ) = 0, U ( F ) = − ( ab ′ − a ′ b ) = − κ , sothat µ = 0. The equiaffine normal (2.33) is W = sgn( κ ) | κ | − / ( a ′ ( u ) ∂ y − b ′ ( u ) ∂ x ). The level setΣ t = { ( u, x, y ) ∈ R : F ( u, x, y ) = t } is the ruled surface given parametrically byΣ t = { ( r, sb ( r ) + κ − ( t − Q ( r )) b ′ ( r ) , − sa ( r ) − κ − ( t − Q ( r )) a ′ ( r )) ∈ R : ( r, s ) ∈ R } . (3.27)By Corollary 2.12, a linear rescaling of (3.27) solves the affine normal flow. Since a and b do notvanish simultaneously, Y = b∂ x − a∂ y never vanishes and is tangent to Σ. Since along Σ there holds0 = F ( u, x, y ) = a ( u ) x + b ( u ) y there exist functions p ( u ) and q ( u ) such that qx = pb and qy = − pa .Note that p and q are necessarily nonvanishing. Then the vector field X = p ( b ′ ∂ x − a ′ ∂ y ) + q∂ z is tangent to Σ and independent of Y , for Y ∧ X = p ( ab ′ − a ′ b ) ∂ x ∧ ∂ y = κp∂ x ∧ ∂ y = 0. Since b ∇ Y W = 0 and b ∇ X W = sgn( κ ) | κ | / qrY , the equiaffine shape operator is given by S ( Y ) = 0 and S ( X ) = − sgn( κ ) | κ | / qrY , so is nilpotent and has zero trace. This shows directly that the surfaceΣ has equiaffine mean curvature zero.Particular special cases of this construction are well known. Taking q = 1 and a ( t ) = sin t and b ( t ) = − cos( t ), so F ( u, x, y ) = x sin u − y cos u , there results the usual helicoid { ( r, s cos r, s sin r ) ∈ R : ( r, s ) ∈ R } . In this case κ = − W = sin u∂ x − cos u∂ y . Taking q = − a ( t ) = e − t and b = − e t , so F ( u, x, y ) = xe − u − ye u , there results the surface x = ye u . In this case κ = 2 and W = 2 − / ( e u ∂ x − e − u ∂ y ). These two examples are given in section 3 of [21].A modification of these examples allows the construction of examples in higher dimensions. Lemma 3.5. Let M be a an open smooth connected submanifold of R n − equipped with the inducedflat affine connection and parallel volume form ω and let B : M → R n be a smooth centroaffineimmersion satisfying (cid:12)(cid:12) B dB (cid:12)(cid:12) = κ for = κ ∈ R , with respect to the standard flat affine connectionand parallel volume form on R n . Let a, b, c ∈ C ∞ ( I ) be smooth functions on a connected opensubinterval I ⊂ R . Endow I × M with the product flat affine connection and the parallel volumeform dt ∧ ω , and regard R n +1 as R n × R , endowed with the standard flat affine connection andparallel volume form. If there is a nonzero constant ¯ κ such that ( − n κ ( ab ′ − a ′ a ) a n − c n − = ¯ κ, (3.28) then the map A : I × M → R n +1 defined by A ( t, u ) = ( a ( t ) B ( uc ( t )) , b ( t )) is a smooth centroaffineimmersion satisfying (cid:12)(cid:12) A dA (cid:12)(cid:12) = ¯ κ . In particular,(1) If a, b ∈ C ∞ ( I ) are linearly independent solutions on I ⊂ R of a second order lineardifferential equation and a does not vanish on I , then the map A : I × M → R n +1 definedby A ( t, u ) = ( a ( t ) B ( u/a ( t )) , b ( t )) is a smooth centroaffine immersion satisfying (cid:12)(cid:12) A dA (cid:12)(cid:12) =( − n τ κ where τ is the constant such that τ = ab ′ − a ′ b .(2) If n is even, and a and ab ′ − a ′ b are nonvanishing on I , then c = a − ( ab ′ − a ′ b ) / ( n − yields a solution of (3.28) with ¯ κ = κ .(3) If n is odd, and a and ab ′ − a ′ b are nonvanishing on I , then c = a − | ab ′ − a ′ b | / ( n − yieldsa solution of (3.28) with ¯ κ = − sgn( ab ′ − a ′ b ) κ .Proof. Although this is a straightforward computation, the details are given for clarity. Using aslightly abusive matrix notation, and calculating the determinant by expansion by cofactors along the last row, one obtains (cid:12)(cid:12) A dA (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) aB a ′ B + ac ′ P n − i =1 u i ∂B∂u i acdBb b ′ (cid:12)(cid:12)(cid:12)(cid:12) = ( − n +1 b (cid:12)(cid:12) a ′ B + ac ′ P n − i =1 u i ∂B∂u i acdB (cid:12)(cid:12) + ( − n b ′ (cid:12)(cid:12) aB acdB (cid:12)(cid:12) = ( − n +1 (cid:0) b (cid:12)(cid:12) a ′ B acdB (cid:12)(cid:12) + b (cid:12)(cid:12) ac ′ P n − i =1 u i ∂B∂u i acdB (cid:12)(cid:12) − b ′ (cid:12)(cid:12) aB acdB (cid:12)(cid:12)(cid:1) = ( − n +1 ( ba ′ − b ′ a ) a n − c n − (cid:12)(cid:12) B dB (cid:12)(cid:12) = ( − n κ ( ab ′ − a ′ a ) b n − c n − , (3.29)where the penultimate equality holds because P n − i =1 u i ∂B∂u i is a linear combination of the columnsof dB . Choosing c ( t ) = 1 /a ( t ) yields the conclusion (1). The remaining conclusions are straight-forward. (cid:3) For example, applying Lemma 3.5 with B being the helicoid and a ( t ) = cosh t and b ( t ) = sinh( t )yields, after relabeling variables, A ( x , x ) = cosh x cos( x sech x )cosh x sin( x sech x )sinh x , (3.30)that satisfies (cid:12)(cid:12) A dA (cid:12)(cid:12) = − R . It follows that, for any Q ∈ C ∞ ( R ), the level sets of thefunction F ( x ,x , x , x , x )= x cosh x cos( x sech x ) + x cosh x sin( x sech x ) + x sinh x + Q ( x , x )(3.31)are equiaffine mean curvature zero smooth hypersurfaces ruled by 2-planes.A different example, directly generalizing the helicoid, can be obtained by starting from a cen-troaffine immersion and explicitly solving for a diffeomorphism as in Theorem 3.4. Look for animmersion A : R → R that takes values in the Euclidean unit sphere. The standard parame-terization of the upper half sphere by spherical polar coordinates does not satisfy that (cid:12)(cid:12) A dA (cid:12)(cid:12) isconstant, but by solving (3.26) there is obtained the parameterization A : Σ → R , A ( x , x ) = p − x cos x p − x sin x x , (3.32)defined on the strip Σ = { x ∈ R : x ∈ ( − , } and satisfying (cid:12)(cid:12) A dA (cid:12)(cid:12) = − 1. There aremany diffeomorphisms from R to Σ with Jacobian determinant 1. Simply take a diffeomorphism ψ : ( − , → R and define ( x , x ) = φ ( t, s ) = ( ψ ( t ) , s/ψ ′ ( t )). For a concrete example, take ψ ( t ) = tanh t . After relabeling t and s , there results the parameterization A : R → R , A ( x , x ) = sech x cos( x cosh x )sech x sin( x cosh x )tanh x , (3.33)that satisfies (cid:12)(cid:12) A dA (cid:12)(cid:12) = − 1. The immersion (3.33) can be obtained from (2) of Lemma 3.5, bytaking a = sech t , b = tanh t and c = cosh t . It follows that the level sets of the function (1.1) areequiaffine mean curvature zero smooth hypersurfaces ruled by 2-planes.The special case of Theorem 3.4 where the image A ( M ) is a graph can be modeled by taking a i +1 ( u , . . . , u n ) = u i for 1 ≤ i ≤ n . Write a ( u ) = f ( u ). Then dF ( V ) = f − P ni =1 u i f i where f i = df ( ∂ u i ). Choose any function g so that P ni =1 u i g i = g , choose κ = 0, and define f = g + κ .Then dF ( V ) = κ , so U ( F ) = κ . The level sets of F passing through points of R in the complementof the set where g fails to be smooth are nondegenerate smooth hypersurfaces having zero equiaffinemean curvature. QUIAFFINE GEOMETRY OF LEVEL SETS 29 To have a concrete example, let k be a positive integer and let g ( u, v ) = ( u k + v k ) / k . Afterrelabeling variables there results F ( x , x , x , x , x ) = ( x k + x k ) / k x + x + x x + x x . (3.34)Note that the level set { x ∈ R : F ( x ) = c } is the graph { x = u ( x , x , x , x ) } of the function u ( x , x , x , x ) = c − x x − x x x k + x k ) / k , (3.35)on { x ∈ R : x = 0 } along ∂ . These level sets are smooth except where both x and x vanish. These examples are somewhat unsatisfying because of the nonregularity along the locus { x ∈ R : x = 0 = x } .4. Comparison of equiaffine and unit normals of a hypersurface in a space form The second fundamental form of an immersion Σ → M with respect to an affine connection b ∇ on M depends only on the the projective equivalence class of b ∇ , as, for vector fields X and Y tangent to Σ, the projections of b ∇ X Y + γ ( X ) Y + Y γ ( X ) and b ∇ X Y onto the normal bundle ofΣ are equal. Thus it makes sense to speak of the second fundamental form of an immersion in amanifold equipped with a projective structure, and it makes sense to say that such an immersionis nondegenerate if the second fundamental form is nondegenerate.From (2.1) it is apparent that the connections induced on Σ by a fixed b ∇ ∈ [ b ∇ ] with respectto transversals N and ˜ N = aN are the same, while the connections ∇ and ˜ ∇ induced on Σ by b ∇ and b ∇ + 2 γ ( i δ j ) k with respect to a fixed transversal N are projectively equivalent, related by˜ ∇ = ∇ + 2 γ ( I δ J ) K . Here, as throughout this section, tensors on a hypersurface Σ are labeled withcapital Latin abstract indices. Consequently, [ b ∇ ] and a line field transverse to Σ determine on Σ aprojective structure [ ∇ ].Although the nondegeneracy of a hypersurface depends only on the projective equivalence class ofthe ambient connection, the affine normal distributions induced by different connections generatingthe same flat projective structure are different. The precise relation is given by Lemma 4.1. Lemma 4.1. Let Σ be a nondegenerate immersed hypersurface in an ( n + 1) -dimensional manifold M equipped with a projectively flat affine connection b ∇ . If W is a local section over the openset U ⊂ Σ of the affine normal distribution determined by b ∇ then the affine normal distributiondetermined by the projectively equivalent connection b ∇ + 2 σ ( i δ j ) k is spanned over U by W + σ ♯ where σ ♯ is any vector field equal to h IP σ P along U , where h is the representative of the secondfundamental form determined by b ∇ and W .Proof. Let h , τ , and ¯ ∇ be determined by b ∇ + 2 σ ( i δ j ) k and W . Let ∇ be the connection induced by b ∇ and W . Then ¯ ∇ = ∇ + 2 σ ( I δ J ) K and τ I = σ I . As ∇ preserves vol h , h P Q ¯ ∇ I h P Q = − n + 1) σ I .By construction the affine normal distribution determined by b ∇ +2 σ ( i δ j ) k is spanned by ˜ W = W + Z where − ( n + 2) Z P h IP = τ I + h P Q ¯ ∇ I h P Q = − ( n + 2) σ I . This proves the claim. (cid:3) Remark 4.1. In Lemma 4.1, the hypothesis that b ∇ be projectively flat is unnecessary; see [7]. ♦ Lemma 4.1 shows that any transversal spans the affine normal distribution determined by someconnection projectively equivalent to a given projectively flat connection. So, while there is amodel of hyperbolic space for which the Levi-Civita connection of the hyperbolic metric is projec-tively equivalent to the standard flat Euclidean connection, the affine normals of a hypersurfacedetermined by these connections are different. This raises the possibility that there are interestingexamples in the affine geometry of hypersurfaces in constant curvature pseudo-Riemannian spaces. The Levi-Civita connection b ∇ of a pseudo-Riemannian manifold ( M, g ) is projectively flat if andonly if g has constant curvature. When the metric g has indefinite signature there are two notionsof nondegeneracy of an immersed co-oriented hypersurface Σ ⊂ M . First, there is the equiaffinenotion of nondegeneracy with respect to b ∇ , necessary to make sense of the co-oriented equiaffinenormal W determined by b ∇ and and the volume form vol g of g , as in Theorem 2.14. Second, thereis the usual pseudo-Riemannian notion of nondegeneracy of Σ, that the restriction of g to Σ benondegenerate, necessary to make sense of the co-oriented unimodular normal field E g -orthogonalto Σ. That E be unimodular means that | g ( E, E ) | = 1. Theorem 4.2 shows that for a hypersurfacenondegenerate in both senses these normals are proportional exactly when the hypersurface hasconstant Gauss-Kronecker curvature. By the Gauss-Kronecker curvature is meant the determinantof the shape operator determined by the Levi-Civita connection of g and the co-oriented unimodularnormal E . Theorem 4.2. Let ( M, g ) be a constant curvature pseudo-Riemannian manifold, and let Σ be aco-oriented immersed hypersurface in M such that Σ is nondegenerate with respect to the the Levi-Civita connection b ∇ of g and the restriction of g to Σ is a pseudo-Riemannian metric. Then theco-oriented equiaffine normal W of Σ determined by b ∇ and the volume form vol g of g and theco-oriented unimodular orthogonal vector field E are proportional if and only if Σ has constantGauss-Kronecker curvature, in which case W is a constant multiple of E and the equiaffine meancurvature of Σ is a constant multiple of the pseudo-Riemannian mean curvature.Proof. Let D , A , and Λ be the induced connection, shape operator, and second fundamental formdetermined by b ∇ with respect to the co-oriented unimodular vector field E g -orthogonal to Σ. Bydefinition, the Gauss-Kronecker curvature of Σ is det A . The connection one-form determined by b ∇ and E is zero, D is the Levi-Civita connection of the restriction of g to Σ, and Λ and A are relatedby g ( AX, Y ) = ǫ Λ( X, Y ) for X and Y tangent to Σ and ǫ = g ( E, E ). By the proof of Theorem2.14, W = a ( E + Z ) where, by (2.60), − ( n + 2) Z P Λ IP = Λ P Q D I Λ P Q = | det Λ | − D I | det Λ | = D I log det A (the last equality because D preserves g ) and a n +2 = | vol Λ / ( ι ( E ) vol g ) | = | det A | .Since Z is tangent to Σ, E is orthogonal to Σ and nonvanishing, and a does not vanish because Σ isassumed nondegenerate, there vanishes W ∧ E = aZ ∧ E if and only if Z = 0. Since − ( n +2) Z P Λ IP = D I log det A , this occurs if and only if det A is constant. (cid:3) By Theorem 4.2 a nondegenerate hypersurface in a pseudo-Riemannian space form that hasconstant pseudo-Riemannian mean curvature zero and nonzero constant Gauss-Kronecker curvaturehas constant equiaffine mean curvature. If the space form is three-dimensional then these conditionsmean that the pseudo-Riemannian principal curvatures of the hypersurface are constant. In [3], B.-Y. Chen showed that if M is a surface with Riemannian mean curvature zero and constant Gausscurvature in a three-dimensional curvature c Riemannian space form, then either M is totallygeodesic or c > M is isometric to an open submanifold of a Clifford torus. The Cliffordtorus is a special case of the following well-known example of a minimal embedding with constantGauss-Kronecker curvature. Let S n ( r ) be the radius r sphere in ( n + 1)-dimensional Euclideanspace. In [4] it is shown that the embedding S m (cid:16)p m/n (cid:17) × S n − m (cid:16)p ( n − m ) /n (cid:17) → S n +1 (1)(4.1)induced by the embedding R m +1 × R n − m +1 → R n +2 is minimal, with m principal curvatures equalto ± p ( n − m ) /m and n − m principal curvatures equal to ∓ p m/ ( n − m ). In particular, thisembedding has constant Gauss-Kronecker curvature. This shows that there are nontrivial examplesof equiaffine mean curvature zero hypersurfaces in spheres. Recall that a hypersurface in a spaceform is isoparameteric if its pseudo-Riemannian principal curvatures are constant. Since such a QUIAFFINE GEOMETRY OF LEVEL SETS 31 hypersurface has constant Gauss-Kronecker curvature, if no pseudo-Riemannian principal curvatureis zero, Theorem 4.2 applies to show that its equiaffine mean curvature is constant. Corollary 4.3. In a pseudo-Riemannian space form, the equiaffine mean curvature of an isopara-metric hypersurface having nonzero Gauss-Kronecker curvature is constant. It is reasonable to ask under what conditions a constant equiaffine mean curvature hypersurfacein a pseudo-Riemannian space form must be isoparametric. The question is very general and hereno attempt is made to review the related literature. The answer must depend on the curvature ofthe ambient space form, as is illustrated by the theorem of B.-Y. Chen mentioned above. References 1. A. Banyaga, Formes-volume sur les vari´et´es `a bord , Enseignement Math. (2) (1974), 127–131.2. A.-L. Cauchy, M´emoire sur les fonctions qui ne peuvent obtenir que deux valeurs ´egales et de signes contrairespar suite des transpositions opres entre les variables qu’elles renferment , Journal de l’´Ecole Polytechnique (1815), 29–112, Oeuvres (2)1.3. B.-Y. Chen, Minimal surfaces with constant Gauss curvature , Proc. Amer. Math. Soc. (1972), 504–508.4. S. S. Chern, M. do Carmo, and S. 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