Classification of Calabi Hypersurfaces with parallel Fubini-Pick form
aa r X i v : . [ m a t h . DG ] J u l CLASSIFICATION OF CALABI HYPERSURFACESWITH PARALLEL FUBINI-PICK FORM
RUIWEI XU MIAOXIN LEI
Abstract:
In this paper, we present the classification of 2 and 3-dimensional Cal-abi hypersurfaces with parallel Fubini-Pick form with respect to the Levi-Civitaconnection of the Calabi metric.
Primary 53A15; Secondary 53C24, 53C42.
Key words: parallel Fubini-Pick form; centroaffine hypersurface; Calabi geometry;Hessian geometry; Tchebychev affine K¨ahler hypersurface.1.
Introduction
In equiaffine differential geometry, the problem of classifying locally strong convexaffine hypersurfaces with parallel Fubini-Pick form (also called cubic form) has beenstudied intensively, from the earlier beginning paper by Bokan-Nomizu-Simon [1],and then [5],[6],[7], to complete classification of Hu-Li-Vrancken [8]. Here we recallresults about the classification of locally strong convex affine hypersurfaces withparallel Fubini-Pick form ∇ A = 0 with respect to the Levi-Civita connection of theaffine Berwald-Blaschke metric. The condition ∇ A = 0 implies that M is an affinehypersphere with constant affine scalar curvature. Thus Theorem 1 of Li and Penn[14] (see also Theorem 3.7 in [12]) can be restated as follows: Theorem 1.1. [14]
Let x : M → A be a locally strongly convex affine surfacewith ∇ A = 0 . Then, up to an affine transformation, x ( M ) lies on the surface x x x = 1 or strongly convex quadric. The classification of 3-dimensional affine hypersurfaces with parallel Fubini-Pickform, due to Dillen and Vrancken [5].
Theorem 1.2. [5]
Let x : M → A be a locally strongly convex affine hypersurfacewith ∇ A = 0 . Then, up to an affine transformation, either x ( M ) is an open partof a locally strongly convex quadric or x ( M ) is an open part of one of the followingtwo hypersurfaces:(i) x x x x = 1 ;(ii) ( x − x − x ) x = 1 . In [8], Hu-Li-Vrancken introduced some typical examples and gave the completeclassification of locally strongly convex affine hypersurfaces of R n +1 with parallelcubic form with respect to the Levi-Civita connection of the affine Berwald-Blaschkemetric. The first author is partially supported by NSFC 11871197 and 11671121.
In centroaffine differential geometry, Cheng-Hu-Moruz [4] obtained a completeclassification of locally strong convex centroaffine hypersurfaces with parallel cubicform. On the other hand, Liu and Wang [16] gave the classification of the centroaffinesurfaces with parallel traceless cubic form relative to the Levi-Civita connection. In[3], Cheng and Hu established a general inequality for locally strongly convex cen-troaffine hypersurfaces in R n +1 involving the norm of the covariant differentiationof both the difference tensor and the Tchebychev vector field. Applying the clas-sification result of [4], Cheng-Hu [3] completely classified locally strongly convexcentroaffine hypersurfaces with parallel traceless difference tensor.A centroaffine hypersurface is said to be Canonical if its Blaschke metric is flatand its Fubini-Pick form is parallel with respect to its Blaschke metric. In [15],Li and Wang classified the Canonical centroaffine hypersurfaces in R n +1 . In thispaper, we first classify the canonical Calabi hypersurfaces in Calabi geometry. As acorollary, we classify Calabi surfaces with parallel Fubini-Pick form with respect tothe Levi-Civita connection of the Calabi metric. Theorem 1.3.
Let f be a smooth strictly convex function on a domain Ω ∈ R n .If its graph M = { ( x, f ( x )) | x ∈ Ω } has a flat Calabi metric and parallel cubic form.Then M is Calabi affine equivalent to an open part of the following hypersurfaces:(i) elliptic paraboloid; or(ii) the hypersurfaces Q ( c , · · · , c r ; n ) , ≤ r ≤ n. The definitions of
Calabi affine equivalent and hypersurfaces Q ( c , · · · , c r ; n ) willbe given in Section 2 (see Definition 2.1 and Example 2.1, respectively). Corollary 1.4.
Let f be a smooth strictly convex function on a domain Ω ∈ R .If its graph M = { ( x, f ( x )) | x ∈ Ω } has parallel cubic form, M is Calabi affineequivalent to an open part of the following surfaces:(i) elliptic paraboloid; or(ii) the surfaces Q ( c , · · · , c r ; 2) , ≤ r ≤ . Motivated by above classification results in equiaffine differential geometry andcentroaffine differential geometry, we present the classification of 3-dimensional Cal-abi hypersurfaces with parallel Fubini-Pick form with respect to the Levi-Civitaconnection of the Calabi metric in Section 4 and Section 5.
Theorem 1.5.
Let f be a smooth strictly convex function on a domain Ω ∈ R .If its graph M = { ( x, f ( x )) | x ∈ Ω } has parallel cubic form. Then M is Calabiaffine equivalent to an open part of one of the following three types of hypersurfaces:(i) elliptic paraboloid; or(ii) the hypersurfaces Q ( c , · · · , c r ; 3) , ≤ r ≤ or(iii) the hypersurface x = − c ln( x − ( x + x )) , where the constant − c is the scalar curvature of M . Remark: (1) In [20], Xu-Li proved that
ALABI HYPERSURFACES WITH PARALLEL FUBINI-PICK FORM 3
Theorem 1.6. [20]
Let M n ( n ≥ be a Calabi complete Tchebychev affine K¨ahlerhypersurface with nonnegative Ricci curvature. Then it must be Calabi affine equiv-alent to either an elliptic paraboloid or one of the hypersurfaces Q ( c , ..., c r ; n ) . Case (iii) of Theorem 1.5 shows that there exists a class of Calabi completeTchebychev affine K¨ahler hypersurfaces with negative Ricci curvature. Thus therestriction on Ricci curvature in Theorem 1.6 is essential.(2) The Euler-Lagrange equation of the volume variation with respect to theCalabi metric can be written as the following fourth order PDE (see [17] or [13])∆ ln det (cid:18) ∂ f∂x i ∂x j (cid:19) = 0 , (1.1)where ∆ is the Laplacian of the Calabi metric G = P ∂ f∂x i ∂x j dx i dx j . Graph hyper-surfaces M n defined by solutions of (1.1) are called affine extremal hypersurfaces .Case (iii) of Theorem 1.5 shows that there exists a class of new Euclidean completeand Calabi complete affine extremal hypersurfaces.2. Preliminaries
Calabi geometry.
In this section, we shall show some basic facts for theCalabi geometry, see [2] or [19]. Let f be a strictly convex C ∞ -function on a domainΩ ∈ R n . Consider the graph hypersurface M n := { ( x i , f ( x i )) | x n +1 = f ( x , · · · , x n ) , ( x , · · · , x n ) ∈ Ω } . (2.1)This Calabi geometry can also be essentially realised as a very special relative affinegeometry of the graph hypersurface (2.1) by choosing the so-called Calabi affinenormalization with Y = (0 , , · · · , t ∈ R n +1 being the fixed relative affine normal vector field which we call the Calabi affinenormal .For the position vector x = ( x , · · · , x n , f ( x , · · · , x n )) we have the decomposition x ij = c kij x ∗ ∂∂x k + f ij Y, (2.2)with respect to the bundle decomposition R n +1 = x ∗ T M n ⊕ R · Y , where the inducedaffine connection c kij ≡
0. It follows that the relative affine metric is nothing but theCalabi metric G = X ∂ f∂x i ∂x j dx i dx j . The Levi-Civita connection with respect to the metric G has the Christoffel symbolsΓ kij = 12 X f kl f ijl , where and hereafter f ijk = ∂ f∂x i ∂x j ∂x k , ( f kl ) = ( f ij ) − . R.W. XU AND M.X. LEI
Then we can rewrite the Gauss structure equation as follows: x ,ij = X A kij x ∗ ∂∂x k + G ij Y. (2.3)The Fubini-Pick tensor (also called cubic form) A ijk and the Weingarten tensorsatisfy A ijk = A lij G kl = − f ijk , B ij = 0 , (2.4)which means A ijk are symmetric in all indexes. Classically, the tangent vector field T := 1 n X G kl G ij A ijk ∂∂x l = − n X f kl f ij f ijk ∂∂x l (2.5)is called the Tchebychev vector field of the hypersurface M n , and the invariantfunction J := 1 n ( n − X G il G jp G kq A ijk A lpq = 14 n ( n − X f il f jp f kq f ijk f lpq (2.6)is named as the relative Pick invariant of M n . As in the report [10], M n is called a Tchebychev affine K¨ahler hypersurface , if the Tchebychev vector field T is parallelwith respect to the Calabi metric G .The Gauss integrability conditions and the
Codazzi equations read R ijkl = X f mh ( A jkm A hil − A ikm A hjl ) , (2.7) A ijk,l = A ijl,k . (2.8)From (2.7) we get the Ricci tensor R ik = X f jl f mh ( A jkm A hil − A ikm A hjl ) . (2.9)Thus, the scalar curvature is given by R = n ( n − J − n | T | . (2.10)Using Ricci identity, (2.7) and (2.8), we have two useful formulas. Lemma 2.1.
For a Calabi hypersurface, the following formulas hold
12 ∆ | T | = X T i,j + X T i T j,ji + X R ij T i T j , (2.11) n ( n − J = X ( A ijk,l ) + X A ijk A lli,jk + X ( R ijkl ) + X R ij A ipq A jpq . (2.12)Let A ( n + 1) be the group of ( n + 1)-dimensional affine transformations on R n +1 .Then A ( n + 1) = GL ( n + 1) ⋉ R n +1 , the semi-direct product of the general lineargroup GL ( n + 1) and the group R n +1 of all the parallel transports on R n +1 . Define SA ( n + 1) = { φ = ( M, b ) ∈ A ( n + 1) = GL ( n + 1) ⋉ R n +1 ; M ( Y ) = Y } (2.13) ALABI HYPERSURFACES WITH PARALLEL FUBINI-PICK FORM 5 where Y = (0 , · · · , , t is the Calabi affine normal. Then the subgroup SA consistsof all the transformations φ of the following type: X : = ( X j , X n +1 ) t ≡ ( X , · · · , X n , X n +1 ) t φ ( X ) := a ij a n +1 j ! X + b, ∀ X ∈ R n +1 , (2.14)for some ( a ij ) ∈ A ( n ), constants a n +1 j ( j = 1 , · · · , n ) and some constant vector b ∈ R n +1 . Clearly, the Calabi metric G is invariant under the action of SA ( n + 1)on the graph hypersurfaces or, equivalently, under the induced action of SA ( n + 1)on the strictly convex functions, which is naturally defined to be the composition ofthe following maps: f ( x i , f ( x i )) (˜ x i , ˜ f (˜ x i )) := φ ( x i , f ( x i )) φ ( f ) := ˜ f , ∀ φ ∈ SA ( n + 1) . Definition 2.1. [20]
Two graph hypersurfaces ( x i , f ( x i )) and (˜ x i , ˜ f (˜ x i )) , definedrespectively in domains Ω , ˜Ω ⊂ R n , are called Calabi-affine equivalent if they differonly by an affine transformation φ ∈ SA ( n + 1) . Accordingly, we have
Definition 2.2. [20]
Two smooth functions f and ˜ f respectively defined on do-mains Ω , ˜Ω ⊂ R n are called affine equivalent ( related with an affine transformation ϕ ∈ A ( n )) if there exist some constants a n +11 , · · · , a n +1 n , b n +1 ∈ R such that ϕ (Ω) = ˜Ω and ˜ f ( ϕ ( x j )) = f ( x j ) + X a n +1 j x j + b n +1 for all ( x j ) ∈ Ω . (2.15)Clearly, the above two definitions are equivalent to each other.2.2. Canonical Calabi hypersurfaces.
A Calabi hypersurface is called canonical if its Fubini-Pick form is parallel with respect to the Levi-Civita connection of theCalabi metric and its Calabi metric G is flat. In [20] Xu and Li introduced a largeclass of new canonical Calabi hypersurfaces, which are denoted by Q ( c , · · · , c r ; n ).It turns out that these new examples are all Euclidean complete and Calabi complete. Example 2.1. [20]
Given the dimension n and let ≤ r ≤ n . For any positivenumbers c , · · · , c r , define Ω c , ··· ,c r ; n = { ( x , · · · , x n ); x > , · · · , x r > } and consider the following smooth functions f ( x , · · · , x n ) ≡ Q ( c , c , · · · , c r ; n )( x , · · · , x n ):= − r X i =1 c i ln x i + n X j = r +1 x j , ( x , · · · , x n ) ∈ Ω c , ··· ,c r ; n . (2.16) R.W. XU AND M.X. LEI Proof of the Theorem 1.3 and Corollary 1.4
Lemma 3.1.
Let { A i } ≤ i ≤ n be real symmetric matrices satisfying A i A j = A j A i , ∀ ≤ i, j ≤ n . Then there exists an orthogonal matrix such that matrices { A i } ≤ i ≤ n can be simultaneously diagonalized. Proof.
As we know the conclusion obviously holds for the case of n = 2. Nowwe assume that it holds for n = k . Namely, there is an orthogonal matrix P suchthat A , · · · , A k can be simultaneously diagonalized: P A i P − = diag ( λ i E n i , · · · , λ is E n is ) , ≤ i ≤ k, (3.1)where λ i , · · · , λ is are different eigenvalues for any fixed i .In the next we will prove the conclusion still holds for the case of n = k + 1. Since A i A k +1 = A k +1 A i , ∀ ≤ i ≤ k, we obtain( P A i P − )( P A k +1 P − ) = ( P A k +1 P − )( P A i P − ) . (3.2)Denote B k +1 := P A k +1 P − , where B k +1 is a real symmetric matrix. From (3.1) and (3.2) we have λ ip B k +1 pq = B k +1 pq λ iq , ∀ p, q, ≤ i ≤ k. (3.3)Fix an arbitrary index i ∈ { , · · · , k } . If λ ip = λ iq for some indices p = q then,by (3.3), it must hold that B k +1 pq = 0. Therefore, for any pair of indices p = q , if B k +1 pq = 0, then it holds that λ ip = λ iq for each i = 1 , · · · , k . Thus we get B k +1 = diag ( B k +1 n k +11 , · · · , B k +1 n k +1 r ) , where B k +1 n k +1 j (1 ≤ j ≤ r ) are real symmetric matrices of order n k +1 j , and, for any fixed1 ≤ j ≤ r and 1 ≤ i ≤ k , the n k +11 + · · · + n k +1 j − +1 th to n k +11 + · · · + n k +1 j th eigenvaluesof A i are equal. Thus there are a set of orthogonal matrices R n k +1 j , ≤ j ≤ r suchthat R n k +1 j B k +1 n k +1 j R − n k +1 j are diagonal matrices. Let R = diag ( R n k +11 , · · · , R n k +1 r ) . Then the real symmetric matrices A , · · · , A k +1 can be simultaneously diagonalizedby orthogonal matrix RP . ✷ Proof of the Theorem 1.3.
The canonical Calabi hypersurface means ∇ A = 0 and R ijkl = 0 . Hence M n locally is a Euclidean space. We choose local coordinates { u , · · · , u n } such that the Calabi metric is given by G = P ( du i ) , and A ijk = const in thiscoordinates. We consider the following two subcases: Case 1. A ijk = 0 , ∀ i, j, k. Obviously, in this case, M n is an open part of ellipticparaboloid. ALABI HYPERSURFACES WITH PARALLEL FUBINI-PICK FORM 7
Case 2.
Otherwise. Let p ∈ M n be a fixed point with coordinates (0 , · · · , e i = ∂∂u i , and e = (0 , · · · , ,
1) on M n . Let { ω i } be the dual frame field of { e i } . Denote A ( k ) := A e k = X A ( k ) ij du i du j ,A ( k ) ij := A ( e i , e j , e k ) ≡ A ijk . By the Gauss integrability conditions (2.7) and the flatness of the metric G , we have X A iml A jmk − X A imk A jml = 0 , (3.4)which means the following matrix equalities:( A ( k ) ij )( A ( l ) ij ) = ( A ( l ) ij )( A ( k ) ij ) , ∀ ≤ k, l ≤ n. By Lemma 3.1, we get that matrices ( A ( k ) ij ) can be simultaneously diagonalized.There exists an orthogonal constant matrix C = ( c ij ) , for any fixed 1 ≤ k ≤ n, suchthat ( A ( k )¯ i ¯ j ) = C ( A ( k ) ij ) C − = diag ( λ k , λ k , · · · , λ kn ) . Here A ( k )¯ i ¯ j = A (¯ e i , ¯ e j , e k ) and ¯ e i = P c ij e j , ≤ i ≤ n, then¯ A ijk := A (¯ e i , ¯ e j , ¯ e k ) = X c kl A (¯ e i , ¯ e j , e l ) = X c kl λ li δ ij . Since the matrices ( ¯ A ijk ) are symmetric in all indexes, we get:¯ A ijk = ( ¯ A iii , ≤ i = j = k ≤ n, , otherwise. (3.5)From dx = ω i e i = ¯ ω i ¯ e i , we can get ¯ ω i = P c ij ω j = P c ij du j , where ( c ij ) denotesthe inverse matrix of ( c ij ) . Let ¯ u i = P c ij u j , ≤ i ≤ n, then (¯ u , · · · , ¯ u n ) are newEuclidean coordinates of M n , such that ∂∂ ¯ u i = ¯ e i , ≤ i ≤ n. Under these newcoordinates, the tensor ¯ A is expressed as (3.5), thus we have: ( d ¯ e i = P ¯ ω ji ¯ e j + d ¯ u i e, ≤ i ≤ n,dx = P d ¯ u i ¯ e i . (3.6)Since the Calabi metric is flat and ∂∂ ¯ u i are orthonormal, we obtain ¯ ω ji = ¯ A ijk d ¯ u k . Assume that x ∈ M n is an arbitrary point with coordinates ( v , · · · , v n ) . We drawa curve connecting p and x ¯ u i ( t ) = v i t, ≤ t ≤ . Along this curve the equations (3.6) become ( d ¯ e i dt = ¯ A iii v i ¯ e i + v i e, ≤ i ≤ n, dxdt = P v i ¯ e i . (3.7)Consider the ordinary differential equation dudt = au + b. R.W. XU AND M.X. LEI
It is easy to find out its solution u ( t ) = ( ( u (0) + ba ) e at − ba , a = 0 ,u (0) + bt, a = 0 . We may assume that ¯ e i (0) = (0 , · · · , , · · · , , ≤ i ≤ n, where 1 is on i-th entryand ¯ A iii ≥ p . By an arrangement, we can get ( ¯ A iii > , ≤ i ≤ r ;¯ A jjj = 0 , r + 1 ≤ j ≤ n, (3.8)where 1 ≤ r ≤ n and r = n means that ¯ A iii > ≤ i ≤ n. Without loss of generality, we assume v i > , ≤ i ≤ n. Solve equations (3.7), weobtain: ¯ e i = exp( ¯ A iii v i t )¯ e i (0) + A iii exp( ¯ A iii v i t ) e − A iii e, ≤ i ≤ r ;¯ e j = ¯ e j (0) + v j te, r + 1 ≤ j ≤ n ; x ( t ) = x (0) + R t v i ¯ e i ( s ) ds. (3.9)Thus x ( t ) = x (0) + r X i =1 A iii [exp( ¯ A iii v i t ) − e i (0) + r X i =1 A iii [exp( ¯ A iii v i t ) − e − r X i =1 A iii v i te + n X j = r +1 v j t ¯ e j (0) + n X j = r +1
12 ( v j ) t e. (3.10)Evaluate (3.10) at t = 1 we have: x i = x i (0) + 1¯ A iii [exp( ¯ A iii v i ) − , ≤ i ≤ r ; x j = x j (0) + v j , r + 1 ≤ j ≤ n ; x n +1 = x n +1 (0) + r X i =1 (cid:18) A iii [exp( ¯ A iii v i ) − − A iii v i (cid:19) + n X j = r +1
12 ( v j ) . (3.11)Inserting x i and x j into x n +1 , we find x n +1 = r X i =1 A iii x i − r X i =1 A iii ln( ¯ A iii x i + 1) + 12 n X j = r +1 ( x j ) . It is easy to find that x n +1 is affine equivalent to x n +1 = − r X i =1 A iii ln x i + n X j = r +1
12 ( x j ) . (3 . ✷ Proof of Corollary 1.4 By ∇ A = 0, the definition of the Tchebychev vector field T and the Pick invariant J , we can get: ∇ T = 0 and J = const. ALABI HYPERSURFACES WITH PARALLEL FUBINI-PICK FORM 9
It follows that | T | = const. Case 1. | T | = 0. It means thatdet( f ij ) = const > , (3 . R ij = A iml A jml − A ijm A mll = A iml A jml . (3 . ∇ A = 0, we have n ( n − J = X ( R ij ) + X ( R ijkl ) . (3.15)It follows that R ijkl = 0. Then, by (2.10), we obtain the relative Pick invariant n ( n − J = R + n | T | = 0 . Thus f is a strictly convex quadratic function. Case 2. | T | = const > . In this case, we can choose an orthonormal frame field { ˜ e , ˜ e } on M with ˜ e = T | T | , where ∇ ˜ e = 0 , since ∇ T = 0 . From the definition ofthe Riemannian curvature tensor, we get R ijkl = 0 . (3 . ✷ The classification of 3-dimension case
Elementary discussions in terms of a typical basis.
Now, we fix a point p ∈ M n . For subsequent purpose, we will review the well known construction of atypical orthonormal basis for T p M n , which was introduced by Ejiri and has beenwidely applied, and proved to be very useful for various situations, see e.g., [7], [18]and [4]. The idea is to construct from the (1 ,
2) tensor A a self adjoint operatorat a point; then one extends the eigenbasis to a local field. Let p ∈ M n and U p M n = { v ∈ T p M n | G ( v, v ) = 1 } . Since M n is locally strong convex, U p M n iscompact. We define a function F on U p M n by F ( v ) = A ( v, v, v ). Then there isan element e ∈ U p M n at which the function F ( v ) attains an absolute maximum,denoted by µ . Then we have the following lemma. For its proof, we refer the readerto [7] or [12]. Lemma 4.1.
There exists an orthonormal basis { e , · · · , e n } of T p M n such thatthe following hold:(i) A ( e , e i , e j ) = µ i δ ij , for i = 1 , · · · , n .(ii) µ ≥ µ i , for i ≥ . If µ = 2 µ i , then A ( e i , e i , e i ) = 0 . Consider the function F ( v ) = A ( v, v, v ) on U p M n . Let e ∈ U p M n be a vector at which F ( v ) attains an absolute maximum A ( ≥ e , · · · , e n such that { e , · · · , e n } form an orthonormal basis of T p M n , which possesses the following properties: G ( e i , e j ) = δ ij , A ij = µ i δ ij , ≤ i, j ≤ n ; µ ≥ µ i and if µ = 2 µ i , then A ( e i , e i , e i ) = 0 for i ≥ . Using ∇ A = 0 and the Ricci identity, for i ≥
2, we have0 = A i, i − A i,i = 2 A p i R p i + A p R pi i = µ i ( µ − µ i )( µ i − µ ) . (4.1)Therefore we have the following lemma. Lemma 4.2.
Let M n be a Calabi hypersurface with parallel Fubini-Pick form.Then, for every point p ∈ M n , there exists an orthonormal basis { e j } ≤ j ≤ n of T p M n (if necessary, we rearrange the order), satisfying A ( e , e j ) = µ j e j , and there existsa number i , ≤ i ≤ n , such that µ = µ = · · · = µ i = 12 µ ; µ i +1 = · · · = µ n = 0 . Therefore, for a strictly convex Calabi hypersurface with parallel Fubini-Pick form,we have to deal with ( n + 1) cases as follows: Case C . µ = 0. Case C . µ > µ = µ = · · · = µ n = 0 . Case C i . µ = µ = · · · = µ i = µ > µ i +1 = · · · = µ n = 0 for 2 ≤ i ≤ n − . Case C n . µ = µ = · · · = µ n = µ > . When working at the point p ∈ M n , we will always assume that an orthonormalbasis is chosen such that Lemma 4.1 is satisfied.4.2. The settlement of the Cases C and C n . Firstly, about the Case C , wehave the following lemma. Lemma 4.3.
If the Case C occurs, then M n is an open part of elliptic paraboloid. Proof. If µ = 0, then A ( v, v, v ) = 0 for any v ∈ U p M n . (4.2)Put v = √ ( e i + e j ) ∈ U p M n in (4.2), then0 = A ( e i , e i , e j ) + A ( e i , e j , e j ) . On the other hand, put v = √ ( e i − e j ) ∈ U p M n in (4.2), then0 = − A ( e i , e i , e j ) + A ( e i , e j , e j ) . Thus we have A ( e i , e i , e j ) = 0 , ∀ i, j. From 0 = A ( e i + e k , e i + e k , e j ), we have A ( e i , e j , e k ) = 0 , ∀ i, j, k. ALABI HYPERSURFACES WITH PARALLEL FUBINI-PICK FORM 11
Therefore J ≡
0, and M n is an open part of elliptic paraboloid. ✷ Secondly, we have the following important observation:
Lemma 4.4.
The Case C n does not occur. Proof.
Assume that this case does occur. For any i ≥ µ i = µ >
0, then A ( e , v, v ) = µ and A ( v, v, v ) = 0 for any v ∈ { e ⊥ } T U p M n . From the proof ofLemma 4.3, we see that A ( e i , e j , e k ) = 0 , ≤ i, j, k ≤ n. Then, for any unit vector v ∈ { e ⊥ } T U p M n , we have A ( e , e ) = µ e , A ( e , v ) = 12 µ v, A ( v, v ) = 12 µ e . (4.3)By ∇ A = 0, we know that the curvature operator of Levi-Civita R and Fubini-Picktensor A satisfy R ( e , v ) A ( v, v ) = 2 A ( R ( e , v ) v, v ) . (4.4)By (4.4), (4.3) and (2.7), we get µ = 0. This contradiction completes the proof ofLemma 4.4. ✷ In the following we only consider 3-dimensional Calabi hypersurfaces with parallelFubini-Pick form. Therefore, we only need to deal with the
Case C and Case C .In sequel of this paper, we are going to discuss these cases separately.4.3. The settlement of the case C .Lemma 4.5. If the Case C occurs, then M is Calabi affine equivalent to anopen part of the hypersurfaces Q ( c , · · · , c r ; 3) , ≤ r ≤ . Proof.
Denote A kij := A ( e i , e j , e k ) ≡ A ijk , and put a := A , b := A , c := A , d := A . By µ = µ , we can further choose e as a unit vector for which the function F ,restricted to { e ⊥ } T U p M , attains its maximum A ≥
0. It follows that A = 0,and A ≥ A . Thus we get( A ij ) = µ , ( A ij ) = a
00 0 b , ( A ij ) = b b c . (4.5)By a direct calculation, we have R = R = b ( b − a ) ,R = R = R = R = 0 . (4.6)By (2.11) and (2.12), it yields0 = R ( T ) + R ( T ) = 19 b ( b − a )[( a + b ) + c ] , (4 . X ( R ijkl ) + R X ( A pq ) + R X ( A pq ) = X ( R ijkl ) + b ( b − a )( a + 3 b + c ) . (4.8)If b >
0, it contradicts to (4.7). If b <
0, it also contradicts to (4.8). Therefore b = 0. By (4.8) we get R ijkl ( p ) = 0. Since the arbitrary of point p , we have theCalabi metric is flat. Combining ∇ A = 0 and Theorem 1.3, one can get the followingclassification results:(1) if a = 0, c = 0, then M is Calabi affine equivalent to an open part of thehypersurface Q ( c ; 3);(2) if a = 0, c = 0, then M is Calabi affine equivalent to an open part of thehypersurface Q ( c , c ; 3);(3) if a = 0, c = 0, then M is Calabi affine equivalent to an open part of thehypersurface Q ( c , c , c ; 3). ✷ classification of case C By µ = 2 µ >
0, we know A = 0. Thus we have( A ij ) = µ µ
00 0 0 , ( A ij ) = µ µ d d b , ( A ij ) = d b b c . (5.1)By (2.9), we obtain R = − µ , R = − µ + b + d − cd, R = b + d − cd,R = − µ b, R = µ d, R =0 . Using ∇ A = 0 and the Ricci identity, we have0 = A , − A , = 2 A p R p + A p R p = 2 b µ . (5.2)0 = A , − A , = 3 A p R p = 3 µ ( d − µ ) . (5.3)0 = A , − A , = A p R p + A p R p + A p R p (5.4)= µ (2 b + 2 d − cd ) . By (5.2), (5.3) and (5.4), we obtain b = 0 , d = µ = 0 , c = 2 d. Thus the Pick invariant and the scalar curvature are J = 73 µ , R = − µ . (5.5)Now put tangent vectors˜ e := √
22 ( e + e ) , ˜ e := √
22 ( − e + e ) , (5.6) ALABI HYPERSURFACES WITH PARALLEL FUBINI-PICK FORM 13 then { ˜ e , e , ˜ e } forms an orthonormal basis of T p M , with respect to which, theFubini-Pick tensor A takes the following form: A (˜ e , ˜ e ) = √ µ ˜ e ; A (˜ e , e ) = √ µ e ; A (˜ e , ˜ e ) = √ µ ˜ e , (5.7)and A ( e , e ) = √ µ ˜ e ; A ( e , ˜ e ) = 0; A (˜ e , ˜ e ) = √ µ ˜ e . By parallel translation along geodesics (with respect to the Levi-Civita connection ∇ ) through p to a normal neighborhood around p , we can extend { ˜ e , e , ˜ e } toobtain a local orthonormal basis { E , E , E } on a neighborhood of p such that A ( E , E ) = √ µ E ; A ( E , E ) = √ µ E ; A ( E , E ) = √ µ E (5.8)holds at every point in a normal neighborhood. Denote by ω ji the connection formwith respect to the orthonormal frame { E i } . By ∇ A = 0, A i,j ω j = dA i − A j i ω j − A j ω ji , and choose i = 3, we have ω = 0 . (5.9)Similar, by A i,j ω j = dA i − A j i ω j − A j ω ji , and choose i = 2, we have ω = 0 . (5.10)Then (5.9) and (5.10) show that E is a parallel vector field with respect to theLevi-Civita connection. Thus, by (5.5), we have R = 12 R = − µ = const. (5.11)By the above these equalities we have the following lemma: Lemma 5.1.
We have(i) ∇ E = 0 ;(ii) h∇ E i E j , E i = 0 , for any i, j = 2 , . This lemma tell us that the distribution by D := { RE } and D := span { E , E } are totally geodesic. Therefore it follows from the de Rham decomposition theorem([9], pp.187) that as a Riemannian manifold, ( M , G ) is locally isometric to a Rie-mannian product R × H ( − µ ), where H ( − µ ) is the hyperbolic plane of constantnegative curvature − µ , and after identification, the local vector field E is tangentto R and D is tangent to H ( − µ ).Denote by x = ( x , x , x , x ) t the position vector of M in A . Using the standardparametrization of the hypersphere model of H ( − µ ), we see that there exists localcoordinates ( y , y , y ) on M , such that the metric is given by G = ( dy ) + ( dy ) + sinh ( √ µ y )( dy ) , (5.12) and E = ∂x∂y , and ∂x∂y , (sinh( √ µ y )) − ∂x∂y , form a G-orthonormal basis. Wemay assume that E = ∂x∂y and (sinh( √ µ y )) E = ∂x∂y . Then a straightforwardcomputation shows that ∇ ∂x∂y ∂x∂y = 0 , (5.13) ∇ ∂x∂y ∂x∂y = ∇ ∂x∂y ∂x∂y = √ µ coth( √ µ y ) ∂x∂y , (5.14) ∇ ∂x∂y ∂x∂y = − √ µ sinh( √ µ y ) cosh( √ µ y ) ∂x∂y . (5.15)Using the definition of A , we get the following system of differential equations, where,in order to simplify the equations, we have put c = √ µ and Y = (0 , , , t . ∂ x∂y ∂y = c ∂x∂y + Y, (5.16) ∂ x∂y ∂y = c ∂x∂y , (5.17) ∂ x∂y ∂y = c ∂x∂y , (5.18) ∂ x∂y ∂y = c ∂x∂y + Y, (5.19) ∂ x∂y ∂y = c coth( cy ) ∂x∂y , (5.20) ∂ x∂y ∂y = c sinh ( cy ) ∂x∂y − c sinh( cy ) cosh( cy ) ∂x∂y + sinh ( cy ) Y. (5.21)To solve the above equations, first we solve its corresponding system of homogeneousequations. ∂ x∂y ∂y = c ∂x∂y , (5.22) ∂ x∂y ∂y = c ∂x∂y , (5.23) ∂ x∂y ∂y = c ∂x∂y , (5.24) ∂ x∂y ∂y = c ∂x∂y , (5.25) ∂ x∂y ∂y = c coth( cy ) ∂x∂y , (5.26) ∂ x∂y ∂y = c sinh ( cy ) ∂x∂y − c sinh( cy ) cosh( cy ) ∂x∂y . (5.27) ALABI HYPERSURFACES WITH PARALLEL FUBINI-PICK FORM 15
From (5.22), we know that there exist vector valued functions P ( y , y ) and P ( y , y )such that x = P ( y , y ) e cy + P ( y , y ) . (5.28)From (5.23) and (5.24) it then follows that the vector function P is independent of y and y . Hence there exists a constant vector A such that P ( y , y ) = A . Next,it follows from (5.25) that P ( y , y ) satisfies that the following differential equation: ∂ P ∂y ∂y = c P . (5.29)Hence we can write P ( y , y ) = Q ( y ) cosh( cy ) + Q ( y ) sinh( cy ) . (5.30)From (5.26), we then deduce that there exists a constant vector A such that Q ( y ) = A . The last formula (5.27) implies there exist constant vectors A and A such that Q ( y ) = A cos( cy ) + A sin( cy ) . (5.31)Therefore the general solution of system (5.22-5.27) are x = e cy ( A cosh( cy ) + [ A cos( cy ) + A sin( cy )] sinh( cy )) + A , (5.32)where A i are constant vectors. On the other hand, we know that¯ x = (cid:16) , , , − y c (cid:17) t is a special solution of equations (5.16-5.21). Therefore the general solutions ofequations (5.16-5.21) are x = e cy { A cosh( cy ) + [ A cos( cy ) + A sin( cy )] sinh( cy ) } + A + ¯ x. (5.33)Since M is nondegenerate, x − A lies linearly full in A . Hence A , A , A and(0 , , ,
1) are linearly independent vectors. Thus there exists an affine transforma-tion φ ∈ SA (4) such that A = (0 , , , t , A = (1 , , , t , A = (0 , , , t , A = (0 , , , t . Then the position vector x = (cid:16) cosh( cy ) e cy , cos( cy ) sinh( cy ) e cy , sin( cy ) sinh( cy ) e cy , − y c (cid:17) t . (5.34)It follows that, up to an affine transformation φ ∈ SA (4), M locally lies on thegraph hypersurface of function x = − c ln( x − ( x + x )) . (5.35)Thus we finally arrive at the following lemma. Lemma 5.2.
If the Case C occurs, then M is Calabi affine equivalent to anopen part of the hypersurface x = − c ln( x − ( x + x )) , where the constant − c is the scalar curvature of M . Combining Lemma 4.3, Lemma 4.4, Lemma 4.5 and Lemma 5.2, we complete theproof of Theorem 1.5. ✷ References [1] N. Bokan, K. Nomizu, U. Simon: Affine hypersurfaces with parallel cubic forms. TˆohokuMath. J. 42(1990), 101-108.[2] E. Calabi: Improper affine hyperspheres of convex type and a generalization of a theorem ofJ¨orgens. Michgan J. Math. 5(1958), 105-126.[3] X. Cheng, Z. Hu: An optimal inequality on locally strongly convex centroaffine hypersurfaces.J. Geom. Anal. 28(2018), 643-655.[4] X. Cheng, Z. Hu, M. Morus: Classification of the locally strongly convex centroaffine hyper-surfaces with parallel cubic form. Results Math. 72(2017), 419-469.[5] F. Dillen, L. Vrancken: 3-dimensional affine hypersurfaces in R with parallel cubic form.Nagoya Math. J. 124(1991),41-53.[6] F. Dillen, L. Vrancken, L. Yaprak: Affine hypersurfaces with parallel cubic form. NagoyaMath. J. 135(1994), 153-164.[7] Z. Hu, H. Li, U. Simon, L. Vrancken: On locally strongly convex affine hypersurfaces withparallel cubic form. Part I. Diff. Geom. Appl. 27(2009), 188-205.[8] Z. Hu, H. Li, L. Vrancken: Locally strongly convex affine hypersurfaces with parallel cubicform. J. Diff. Geom. 87(2011), 239-307.[9] S. Kobayashi, K. Nomizu: Foundations of Differential Geometry, vol. I, Interscience Publish-ers, New York, 1963.[10] A.-M. Li: Affine K¨ahler manifolds. Report on international congress in Banach Center ofPoland, 2005.[11] A.-M. Li, H. Li, U. Simon: Centroaffine Bernstein problems. Diff. Geom. Appl. 20(2004), no.3, 331-356.[12] A.-M. Li, U. Simon, G. Zhao, Z. Hu: Global Affine Differential Geometry of Hypersurfaces.Second revised and extended edition. De Gruyter Expositions in Mathematics, 11. De Gruyter,Berlin, 2015.[13] A.-M. Li, R.W. Xu, U. Simon, F. Jia: Affine Bernstein Problems and Monge-Amp`ere Equa-tions. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2010.[14] A.-M. Li, G. Penn: Uniqueness theorems in affine differential geometry. Part II. Results Math.13(1988), 308-317.[15] A.-M. Li, C.P. Wang: Canonical centroaffine hypersurfaces in R n +1 . Affine differential geom-etry (Oberwolfach, 1991). Results Math. 20(1991), no. 3-4, 660-681.[16] H.L. Liu, C.P. Wang: Centroaffine surfaces with parallel traceless cubic form. Bull. Belg.Math. Soc. 4(1997), 493-499.[17] H. Li: Variational problems and PDEs in affine differential geometry. PDEs, submanifoldsand affine differential geometry, 9-41, Banach Center Publ., 69, Polish Acad. Sci. Inst. Math.,Warsaw, 2005.[18] H. Li, L. Vrancken: A basic inequality and new characterization of Whitney spheres in acomplex space form. Israel J. Math. 146 (2005), 223-242.[19] A.V. Pogorelov: The Minkowski Multidimensional Problem. John Wiley & Sons, New York -London - Toronto, 1978.[20] R.W. Xu, X.X. Li: On the complete solutions to the Tchebychev-Affine-K¨ahler equation andits geometric significance. Preprint in 2019.. Affine differential geom-etry (Oberwolfach, 1991). Results Math. 20(1991), no. 3-4, 660-681.[16] H.L. Liu, C.P. Wang: Centroaffine surfaces with parallel traceless cubic form. Bull. Belg.Math. Soc. 4(1997), 493-499.[17] H. Li: Variational problems and PDEs in affine differential geometry. PDEs, submanifoldsand affine differential geometry, 9-41, Banach Center Publ., 69, Polish Acad. Sci. Inst. Math.,Warsaw, 2005.[18] H. Li, L. Vrancken: A basic inequality and new characterization of Whitney spheres in acomplex space form. Israel J. Math. 146 (2005), 223-242.[19] A.V. Pogorelov: The Minkowski Multidimensional Problem. John Wiley & Sons, New York -London - Toronto, 1978.[20] R.W. Xu, X.X. Li: On the complete solutions to the Tchebychev-Affine-K¨ahler equation andits geometric significance. Preprint in 2019.