Virtual immersions and minimal hypersurfaces in compact symmetric spaces
aa r X i v : . [ m a t h . DG ] J un VIRTUAL IMMERSIONS AND MINIMAL HYPERSURFACES INCOMPACT SYMMETRIC SPACES
RICARDO A. E. MENDES ∗ † AND MARCO RADESCHI ∗ Abstract.
We show that closed, immersed, minimal hypersurfaces in a com-pact symmetric space satisfy a lower bound on the index plus nullity, whichdepends linearly on their first Betti number. Moreover, if either the minimalhypersurface satisfies a certain genericity condition, or if the ambient space isa product of two CROSSes, we improve this to a lower bound on the indexalone, which is affine in the first Betti number. To prove these, we introducea generalization of isometric immersions in Euclidean space. Compact sym-metric spaces admit (and in fact are characterized by) such a structure with skew-symmetric second fundamental form. Introduction
Let (
M, g ) be a Riemannian manifold, and Σ a minimal immersed submanifold.This means that the second fundamental form of Σ is traceless, or, equivalently,that Σ is a critical point of the area functional. Then one is naturally led to considervariations up to second order, and to define the (Morse) index of Σ as the dimensionof the space of negative variations. When Σ is closed, the index is finite.Many authors have developed methods to produce minimal submanifolds, includ-ing Min-Max Theory (see [Cod14, Nev14] for surveys), desingularization (see forexample [Kap11, CS16]), and equivariant methods (see for example [HL71, Hsi83,Hsi87]). For some of these the index of the minimal submanifold is controlled, whilefor others the topology is controlled. On the other hand, the set of all minimal sub-manifolds of bounded index, area, or topology is the object of active research, inparticular compactness results such as [CS85, CKM15, Sha17] have been obtained.Therefore it is natural to ask how the topology and the index of minimal subman-ifolds are related. One conjecture that fits in this framework is: (see [Nev14, page16], or [ACS16, page 3] for a slightly different formulation)
Conjecture (Marques-Neves-Schoen) . Let ( M, g ) be a compact manifold with pos-itive Ricci curvature, and dimension at least three. Then there exists C > suchthat, for all closed embedded orientable minimal hypersurfaces Σ → M , ind(Σ) ≥ Cb (Σ) where b (Σ) denotes the first Betti number of Σ with real coefficients. Variations of this conjecture include replacing the assumption that the Riccicurvature is positive with other notions of positivity (or non-negativity) of the
Mathematics Subject Classification.
Key words and phrases. minimal hypersurface, compact symmetric space, isometric immersion. ∗ received support from SFB 878: Groups, Geometry & Actions . † received support from DFG ME 4801/1-1. curvature; replacing the index with the extended index ind , that is, the sum ofindex and nullity; and replacing the linear bound with an affine bound of the formind ≥ C ( b − D ).Some special cases of the conjecture above (or variations thereof) have beenrecently established. For example, Ros has considered the case where ( M, g ) is aflat 3-torus, and has found affine bounds on the index — see Theorem 16 in [Ros06](see also [CM16]). The authors of [ACS17] have extended Ros’ work to the casewhere the ambient space M is a flat torus of arbitrary dimension. Namely, theyprovide an affine bound for the index of minimal hypersurfaces which (if the torushas dimension >
4) are required to have points where all principal curvatures aredistinct. Savo [Sav10] has given linear bounds on the index of minimal hypersurfacesin round spheres, and [ACS16] have extended these bounds to the other compactrank one symmetric spaces. Moreover, the methods in [ACS16] sometimes allowfor small perturbations of the ambient metric in certain directions.Note that the results mentioned above mostly apply to ambient spaces in sub-classes of compact symmetric spaces. Our main result applies uniformly to thiswhole class:
Theorem A.
Let ( M, g ) be a compact symmetric space, G its isometry group, and Σ ⊂ M a closed, immersed minimal hypersurface. Then the extended index of Σ satisfies ind (Σ) ≥ (cid:18) dim G (cid:19) − b (Σ) . To prove Theorem A (as well as the previous results mentioned above) one needsto produce enough negative variations, and roughly speaking, these come from co-ordinates of vector fields. In [Ros06], [ACS17] about flat tori, the tangent bundle istrivial, and a choice of parallelization leads to such coordinates. In [Sav10], [ACS16],such coordinates come from an embedding of the ambient manifold (
M, g ) into Eu-clidean space, an idea that goes back at least to [Sim68] (see also [Sav10, Corollary2.2]). Our method of proof generalizes all of these: we consider embeddings of thetangent bundle of M into a flat trivial bundle M × V over M , such that the naturalflat connection on M × V induces the Levi-Civita connection of M .Such structures, which we call virtual immersions , exhibit an extrinsic geometrysimilar to the classical case. More precisely, one may define the normal bundle,second fundamental form, and normal connection, and these satisfy identities anal-ogous to the fundamental equations of Gauss, Codazzi, and Ricci. The importantdifference is that the second fundamental form is not necessarily symmetric, andin fact the case where it is symmetric corresponds exactly to classical isometricimmersions into Euclidean space. In the present article, we mostly consider theopposite extreme, namely virtual immersions with skew-symmetric second funda-mental form. We show that every compact symmetric space admits a natural suchvirtual immersion, which lies at the heart of the proof of Theorem A.By the Nash Embedding Theorem, every Riemannian manifold admits an iso-metric embedding into Euclidean space. In contrast, virtual immersions with skew-symmetric second fundamental form are extremely rigid, and in fact their existence characterizes symmetric spaces: IRTUAL IMMERSIONS AND MINIMAL HYPERSURFACES 3
Theorem B.
Let ( M, g ) be a compact Riemannian manifold. It admits a virtualimmersion Ω with skew-symmetric second fundamental form if and only if it is asymmetric space. In this case, Ω is essentially unique. Let (
M, g ) be a compact symmetric space. In some situations, one may “im-prove” Theorem A to obtain linear or affine bounds on the index, instead of theextended index, of closed immersed minimal hypersurfaces in M . For example when M is a CROSS, we recover, in a uniform way, linear bounds for the index, althoughwith worse constants than the ones obtained in [ACS16] — see Corollary 19. Inhigher rank, we have: Theorem C.
Let M = G/H be a compact symmetric space of rank r ≥ , with G = Isom( M ) , and Σ ⊂ M a closed, immersed minimal hypersurface. Then anaffine bound of the form ind(Σ) ≥ (cid:18) dim G (cid:19) − ( b (Σ) − D ) holds in the following cases:a) the hypersurface Σ contains a point where all principal curvatures are distinct,and D = 2 r − z ( h ) . Here h denotes the Lie algebra of H , and z ( h ) itscenter.b) M is a product of two CROSSes M = M × M , and D is one plus the numberof two-dimensional factors. Both Theorem C and Corollary 19 are special cases of a more general, albeittechnical, result — see Theorem 18.Part (a) of Theorem C generalizes the main result of [ACS17] from tori to com-pact symmetric spaces. Part (b) may be compared with [ACS16, Theorems 10,11],which provide a linear bound for the index of closed minimal hypersurfaces of prod-ucts of two spheres S a × S b with ( a, b ) = (2 , ≥ C ( b − D ) trivially implies the linear bound ind ≥ C CD b . One situation where Σ isnecessarily unstable is when M has positive Ricci curvature and Σ is two-sided (forexample when both M and Σ are orientable). In particular, we have: Corollary D.
Let M be an orientable compact symmetric space whose universalcover has no Euclidean factors. Then the conclusion of the Marques-Neves-SchoenConjecture holds if M is a product of two CROSSes, or if Σ has a point where theprincipal curvatures are distinct. Acknowledgements.
It is a pleasure to thank Lucas Ambrozio for many enlight-ening discussions during this project, especially regarding Lemma 14 and TheoremC(b). The final part of this project was carried out while the second-named au-thor visited the University of Cologne. The second-named author wishes to thankAlexander Lytchak for his hospitality during the visit.
Conventions.
We will denote by R the curvature tensor, and follow the sign con-vention in [dC92, page 89]. Namely, R ( X, Y ) Z = ∇ Y ∇ X Z − ∇ X ∇ Y Z + ∇ [ X,Y ] Z Shape operators will be defined as in [dC92, page 128], that is, S η ( X ) = − ( ∇ X η ) T R. MENDES AND M. RADESCHI Virtual immersions and their fundamental equations
Let (
M, g ) be a compact Riemannian manifold. We define a generalization ofisometric immersions of (
M, g ) into Euclidean space. Namely, we consider an iso-metric embedding of
T M into a trivial bundle M × V , such that the natural (flat)connection D of M × V induces the Levi-Civita connection ∇ on T M . To makecomputations more convenient, we phrase this definition in the following, slightlydifferent way — see Proposition 4 for a proof that these two definitions coincide.
Definition 1.
Let (
M, g ) be a Riemannian manifold, and V a finite-dimensionalreal vector space endowed with an inner product h , i . Let Ω be a V -valued one-formon M . We say Ω is a virtual immersion if the following two conditions are satisfied:a) h Ω p ( X ) , Ω p ( Y ) i = g p ( X, Y ) for every p ∈ M , and every X, Y ∈ T p M .b) h ( d Ω) p ( X, Y ) , Ω p ( Z ) i = 0 for every p ∈ M , and every X, Y, Z ∈ T p M .We say two virtual immersions Ω i : T M → V i , i = 1 , V , h , i ) → ( V , h , i ) making the obvious diagram commute. Example . Let ψ : ( M, g ) → V be an isometric immersion. Then Ω = dψ is avirtual immersion in the above sense. Example . Let Ω i : T M → V i be virtual immersions, for i = 1 ,
2, and let a , a ∈ C ∞ ( M ) such that a + a = 1 everywhere on M . Then the map Ω ⊕ Ω : T M → V ⊕ V given by v ( a Ω ( v ) , b Ω ( v )) is again a virtual immersion. This followsfrom a straight-forward computation.Given a virtual immersion Ω, we shall identify T M with the image of the map( p, v ) ( p, Ω p ( v )) in M × V .Condition (a) in Definition 1 yields a decomposition of the trivial vector bundle M × V as a direct sum M × V = T M ⊕ νM of T M ⊂ M × V and its orthogonalcomplement, the normal bundle νM . Given ( p, X ) ∈ M × V , we shall write X = X T + X ⊥ for the decomposition into the tangent and normal parts.The natural connection D on M × V induces connections D T (respectively D ⊥ ,the normal connection ) on T M (resp. νM ), given by D TX Y = ( D X Y ) T (resp. D ⊥ X η = ( D X η ) ⊥ ). Here X, Y are vector fields on M , while η is a section of thenormal bundle. Proposition 4.
Let Ω be a V -valued one-form on M satisfying condition (a) inDefinition 1. Then, condition (b) is equivalent to D T = ∇ .Proof. Recall that(1) d Ω( X, Y ) = D X Y − D Y X − [ X, Y ]so that taking the tangent part yields d Ω( X, Y ) T = D TX Y − D TY X − [ X, Y ] . Condition (a) implies that D T is compatible with the metric g , and by the aboveformula Condition (b) is equivalent to D T being torsion-free. Since these twoproperties characterize the Levi-Civita connection, the result follows. (cid:3) Definition 5.
Let Ω be a V -valued virtual immersion, X, Y be smooth vector fieldson M , and η a smooth section of νM . Define the second fundamental form of Ω by II ( X, Y ) = ( D X Y ) ⊥ IRTUAL IMMERSIONS AND MINIMAL HYPERSURFACES 5 and the shape operator in the direction of a normal vector η by S η ( X ) = − ( D X η ) T . Note that the second fundamental form and the shape operator are tensors. Inview of Proposition 4, we may write D X Y = ∇ X Y + II ( X, Y )(2) D X η = − S η X + D ⊥ X η (3) Remark . If Ω is a virtual immersion, then its second fundamental form is sym-metric if and only if d Ω = 0, or, equivalently, Ω locally comes from an isometricimmersion of M into Euclidean space. Indeed, by (1), the normal part of d Ω equals II ( X, Y ) − II ( Y, X ).The fundamental equations of the extrinsic geometry of submanifolds of Eu-clidean space carry over in similar form to virtual immersions. In fact, followingalmost verbatim the computations in, for example, [dC92, Ch. 6.3], one gets thefollowing.
Proposition 7.
Let Ω be a virtual immersion of the Riemannian manifold ( M, g ) with values in V . Then the following identities hold:a) Weingarten’s equation h S η ( X ) , Y i = h II ( X, Y ) , η i b) Gauss’ equation R ( X, Y, Z, W ) = h II ( Y, W ) , II ( X, Z ) i − h II ( X, W ) , II ( Y, Z ) i c) Ricci’s equation (cid:10) R ⊥ ( X, Y ) η, ζ (cid:11) = − (cid:10) ( S tη S ζ − S tζ S η ) X, Y (cid:11) d) Codazzi’s equation h ( D X II )( Y, Z ) , η i = h ( D Y II )( X, Z ) , η i . Index of minimal hypersurfaces
In this section we show that the method of proof used in [Sav10, ACS16, ACS17]applies not only to immersions of the ambient manifold M into Euclidean space,but also to virtual immersions. The statements that we need, along with theirproofs, are essentially the same as in the classical case. We include them here forthe sake of completeness and to fix notations.Let ( M, g ) be a Riemannian manifold, and Σ → M be a closed minimal immersedhypersurface. Recall that the Jacobi operator J Σ is the self-adjoint operator on thespace Γ( ν Σ) of sections of the normal bundle of Σ, and it is defined by: J Σ ( X ) = ∆ ⊥ X + (cid:0) | A | + Ric M ( N, N ) (cid:1) X where ∆ ⊥ is the normal Laplacian and A is the second fundamental form of theimmersion Σ → M . The Morse index of Σ (resp. the nullity of Σ) is the index(resp. the dimension of the kernel) of the quadratic form Q ( X, X ) = − Z Σ J Σ ( X ) · X = Z Σ |∇ ⊥ X | − (cid:0) | A | + Ric M ( N, N ) (cid:1) | X | . The next lemma is equivalent to [ACS16, Proposition 3] (see also [Sav10] and[Ros06, Theorem 16]).
R. MENDES AND M. RADESCHI
Lemma 8.
Suppose H is a vector space of dimension b , and let X : H → Γ( ν Σ) ℓ , X ( ω ) = ( X ( ω ) , . . . X ℓ ( ω )) be a linear map such that for all ω ∈ H , (4) ℓ X i =1 Q ( X i ( ω ) , X i ( ω )) ≤ resp. < . Then ind (Σ) ≥ ℓ b (resp. ind(Σ) ≥ ℓ b ).Proof. Let E m ⊂ Γ( ν Σ) be the sum of eigenspaces of J Σ with non-positive (resp.negative) eigenvalue, and letΦ : H →
Hom( E, R ℓ ) , ω (cid:16) Y (cid:0) h Y, X ( ω ) i L , . . . h Y, X ℓ ( ω ) i L (cid:1)(cid:17) where h X, Y i L := R Σ h X, Y i . Notice that m = ind (Σ) (resp. m = ind(Σ)) and ineither case one must prove b ≤ ℓm .By contradiction, if b > ℓm = dim Hom( E, R ℓ ), then by dimension reasonsker Φ = 0 and, given ω ∈ ker Φ nonzero, it follows that X i ( ω ) ⊥ E for all i = 1 , . . . ℓ .Therefore Q ( X i ( ω ) , X i ( ω )) > Q ( X i ( ω ) , X i ( ω )) ≥
0) and, taking the sumover all i = 1 , . . . ℓ one gets the desired contradiction with equation (4). (cid:3) Suppose M is endowed with a virtual immersion Ω : T M → V , with secondfundamental form II . For any point p ∈ M and vectors x, y ∈ T p M , defineACS( x, y ) := | y | tr (cid:16) | II ( · , x ) | − R ( · , x, · , x ) (cid:17) + | x | tr (cid:16) | II ( · , y ) | − R ( · , y, · , y ) (cid:17) (5) − (cid:16) | II ( x, y ) | − R ( x, y, x, y ) (cid:17) − | II ( y, y ) | This quantity appears naturally in the proof of Proposition 9, more specifically inequation (7).The next result is equivalent (in the case of classical immersions) to Proposition2 in [ACS16]:
Proposition 9.
Suppose M admits a virtual immersion Ω :
T M → V , dim V = d ,such that, for every point p ∈ M and every x, y ∈ T p M orthonormal vectors, ACS ( x, y ) ≤ (resp. < ).Then, for every closed minimal immersed hypersurface Σ → M , ind (Σ) ≥ (cid:18) d (cid:19) − b (Σ) resp. ind(Σ) ≥ (cid:18) d (cid:19) − b (Σ) ! . Proof.
Let Σ → M be a closed, immersed, minimal hypersurface. Also, let θ , . . . θ d denote an orthonormal basis of V .Locally around every point of Σ, it is possible to choose a unit normal vector N ,which is unique up to sign. Given indices 1 ≤ i < j ≤ d and a harmonic 1-form ω on Σ, let ω denote the vector field on Σ such that h ω , Y i = ω ( Y ) for any vectorfield Y in Σ, and define(6) X ij ( ω ) := (cid:10) ω ∧ N, θ i ∧ θ j (cid:11) N. Notice that the definition of X ij ( ω ) does not depend on the specific choice of unitnormal vector N , and therefore it defines a global section of ν Σ, even when Σ is1-sided and there is no global unit vector field N defined on the whole of Σ. IRTUAL IMMERSIONS AND MINIMAL HYPERSURFACES 7
Letting H denote the space of harmonic 1-forms on Σ and letting ℓ = (cid:0) d (cid:1) , thisdefines a linear map X : H → Γ( ν Σ) ℓ , X ( ω ) = (cid:0) X ij ( ω ) (cid:1) i,j . The idea is to apply Lemma 8 to get the result. In order to do this, we mustcompute P i Suppose M admits a virtual immersion Ω : T M → V , dim V = d , such that for every point p ∈ M and every x, y ∈ T p M orthonormal vectors, ACS ( x, y ) ≤ . Let Σ → M be a closed minimal immersed hypersurface, andlet D denote the dimension of the space of harmonic one-forms ω on Σ such that J Σ ( X ij ( ω )) = 0 for all i, j , where X ij ( ω ) is defined in (6) . Then ind(Σ) ≥ (cid:18) d (cid:19) − ( b (Σ) − D ) . Proof. This proof is similar to the proofs of Lemma 8 and Proposition 9, so we usethe same notations and only indicate the necessary modifications.Let H be the space of harmonic 1-forms on Σ, and H ′ ⊂ H the orthogonalcomplement to the space of harmonic 1-forms ω such that J Σ ( X ij ( ω )) = 0 for all i, j . Thus dim( H ′ ) = b − D .Let m = ind(Σ), ℓ = (cid:0) d (cid:1) and consider the restriction of Φ : H → Hom( E, R ℓ )to H ′ , where E denotes the space spanned by the eigenfunctions of J Σ associatedto negative eigenvalues. Assuming for a contradiction that b − D > ℓm yields a R. MENDES AND M. RADESCHI non-zero ω ∈ H ′ such that Φ( ω ) = 0. Then Q ( X ij ( ω ) , X ij ( ω )) ≥ i, j ,and, since ACS ≤ 0, we have Q ( X ij ( ω ) , X ij ( ω )) = 0 for all i, j . But X ij ( ω ) is alinear combination of eigenfunctions with non-negative eigenvalues, so J Σ ( X ij ( ω ))vanishes identically for all i, j , a contradiction. (cid:3) Skew-symmetric second fundamental form In this section we define a natural virtual immersion with skew-symmetric secondfundamental form associated to any compact symmetric space, and use it to proveTheorem A. Then we show that this is in fact the unique example of a virtualimmersion with skew-symmetric second fundamental form, thus proving TheoremB. We start by fixing some notations (see for example [Bes08, Chapter 7] for generalinformation about symmetric spaces). Let ( M, g ) be a compact symmetric space,and p ∈ M . Choose a closed transitive subgroup G of the isometry group of M suchthat ( G, H ) is a symmetric pair, where H = G p . Denote by π : G → G/H = M the natural projection g J g K = gp . Choose an Ad G -invariant metric h , i onthe Lie algebra g such that π is a Riemannian submersion, and let m ⊂ g be theorthogonal complement of h with respect to this metric. Then m is isometric to T p M via the differential of π at the identity e ∈ G ; h and m are Ad H -invariant;and they satisfy [ m , m ] = h .Define G × H m as the quotient of G × m by the action of H given by h. ( g, X ) =( gh − , Ad h X ), and denote by J g, X K the image of ( g, X ) ∈ G × m under the quotientmap. G × H m comes with a natural action by G , defined by g ′ . J g, X K = J g ′ g, X K .Identify the tangent bundle T M with G × H m by extending the isomorphism m → T p M to the G -equivariant isomorphism J g, X K dg ( X ) . With this identification, we define a g -valued one-form Ω on M by(8) Ω ( J g, X K ) = Ad g X. Lemma 11. The g -valued one-form Ω defined in Equation (8) is a virtual immer-sion. At J g K ∈ M , the tangent and normal spaces are Ad g m and Ad g h , respectively.The second fundamental form is given by II (cid:0) J g, X K , J g, Y K (cid:1) = Ad g ([ X, Y ]) and the curvature of the normal connection is given by R ⊥ ( X, Y ) η = [[ X, Y ] , η ] . Proof. It is clear from (8) that the tangent and normal spaces are Ad g m and Ad g h .Let X ∈ g . Under the identification of T M with G × H m that we are using, theaction field X ∗ is given by X ∗ J g K = J g, (Ad g − X ) m K . Indeed, dg − (cid:18) ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 J e tX g K (cid:19) = ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 J g − e tX g K = dπ e (Ad g − X ) = (Ad g − X ) m . IRTUAL IMMERSIONS AND MINIMAL HYPERSURFACES 9 Given X, Y ∈ g , we then have D X ∗ Ω ( Y ∗ ) = ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 Ad e tX g ((Ad g − e − tX Y ) m )= Ad g (cid:0) [Ad g − X, (Ad g − Y ) m ] − (Ad g − [ X, Y ]) m (cid:1) By G -equivariance, it is enough to show that, for every X, Y ∈ m , we have d Ω ( X ∗ , Y ∗ ) Tp = 0 and II ( X, Y ) p = [ X, Y ]. Plugging g = e in the equation above,and using the fact that [ m , m ] = h , we have D X ∗ Ω ( Y ∗ ) = [ X, Y ] . The tangent part of this is zero, so that d Ω ( X ∗ , Y ∗ ) Tp = D X ∗ Ω ( Y ∗ ) Tp − D Y ∗ Ω ( X ∗ ) Tp − Ω ([ X ∗ , Y ∗ ]) p = 0 − − is a virtual immersion.Moreover, II ( X, Y ) p = D X ∗ Ω ( Y ∗ ) ⊥ p = [ X, Y ].Finally, we compute the curvature of the normal connection using Ricci’s equa-tion (see Proposition 7(c)). From the equation for the second fundamental formabove, we see that the shape operator is given by S η ( X ) = − [ X, η ]. Therefore (cid:10) R ⊥ ( X, Y ) η, ξ (cid:11) = h [ S η , S ξ ] X, Y i = h [[ X, ξ ] , η ] − [[ X, η ] , ξ ] , Y i = h− [[ ξ, η ] , X ] , Y i = h [ Y, X ] , [ ξ, η ] i = h [[ X, Y ] , η ] , ξ i where we have used the Jacobi identity in the third equal sign, and bi-invarianceof h , i in the last two equal signs. (cid:3) Example . Let M = S n − , the unit ( n − n ),with Lie algebra so ( n ). The latter may be identified with ∧ R n via the formula x ∧ y xy t − yx t , where x, y are viewed as column n -vectors. Take the base point p to be the first standard basis vector (1 , , . . . , t ∈ R n . Then the virtual immersionΩ : T M → so ( n ) defined in (8) is simply given by ( p, v ) p ∧ v . To prove this,one notes that the map given by this formula and Ω are both O( n )-equivariant,and that they coincide at the point p ∈ M . Remark . Geometrically, we may think of the Lie algebra g as the space ofKilling fields on M . Then, the map Ω defined in (8) sends the tangent vector J g, X K ∈ T J g K M to the unique Killing field with this value at J g K ∈ M , and zerocovariant derivative at this point. Equivalently, Ω ( J g, X K ) is the Killing field ofsmallest norm (as an element of g ) that has the value J g, X K at J g K . Indeed, thisfollows from the formula X ∗ J g K = J g, (Ad g − X ) m K . Proof of Theorem A. Let ( M, g ) be a compact symmetric space, and consider Ω defined in Equation (8). By Lemma 11, this is a virtual immersion with skew-symmetric second fundamental form. By the Gauss equation (Proposition 7(b)), R ( X, Y, X, Y ) = | II ( X, Y ) | , so that in particular the ACS quantity defined inEquation (7) vanishes identically. Now the result follows from Proposition 9. (cid:3) Next we proceed to the proof of Theorem B. We need the following two lemmas. Lemma 14. Let ( M, g ) be a compact Riemannian manifold, and Ω a V -valuedvirtual immersion with skew-symmetric second fundamental form II . Then:a) h R ( X, Y ) Z, W i = h II ( X, Y ) , II ( Z, W ) i . b) ( D X II )( Y, Z ) = − R ( Y, Z ) X .c) ∇ R = 0 . In particular, ( M, g ) is a locally symmetric space.Proof. a) Start with Gauss’ equation (see Proposition 7(b)), R ( X, Y, Z, W ) = h II ( Y, W ) , II ( X, Z ) i − h II ( X, W ) , II ( Y, Z ) i Applying the first Bianchi identity yields0 = − (cid:0) h II ( X, Y ) , II ( Z, W ) i + h II ( Y, Z ) , II ( X, W ) i + h II ( Z, X ) , II ( Y, W ) i (cid:1) so that using Gauss’ equation one more time we arrive at h R ( X, Y ) Z, W i = h II ( X, Y ) , II ( Z, W ) i . b) First we argue that ( D X II )( Y, Z ) is tangent. Indeed, for any normal vector η ,Codazzi’s equation (Proposition 7(d)) says that h ( D X II )( Y, Z ) , η i = h ( D Y II )( X, Z ) , η i . Thus the trilinear map ( X, Y, Z ) 7→ h ( D X II )( Y, Z ) , η i is symmetric in the firsttwo entries and skew-symmetric in the last two entries, which forces it to vanish.Next we let W be any tangent vector and compute h ( D X II )( Y, Z ) , W i = h D X ( II ( Y, Z )) , W i = − h II ( Y, Z ) , D X W i = − h II ( Y, Z ) , II ( X, W ) i = − h R ( Y, Z ) X, W i where in the last equality follows we have used part (a).c) Since the natural connection D on M × V is flat, it follows that for any vectorfields X, Y, Z, W , we have0 = D X ( D Y ( II ( Z, W ))) − D Y ( D X ( II ( Z, W ))) − D [ X,Y ] ( II ( Z, W )) . Fix p ∈ M , and take vector fields such that [ X, Y ] = 0 and ∇ Z = ∇ W = 0 at p ∈ M . Then, evaluating the equation above at p ∈ M , we have0 = D X (cid:0) ( D Y II )( Z, W ) + II ( ∇ Y Z, W ) + II ( Z, ∇ Y W ) (cid:1) − D Y (cid:0) ( D X II )( Z, W ) + II ( ∇ X Z, W ) + II ( Z, ∇ X W ) (cid:1) = D X ( − R ( Z, W ) Y ) + II ( ∇ X ∇ Y Z, W ) + II ( Z, ∇ X ∇ Y W ) − D Y ( − R ( Z, W ) X ) − II ( ∇ Y ∇ X Z, W ) − II ( Z, ∇ Y ∇ X W )= − ( D X R )( Z, W ) Y + ( D Y R )( Z, W ) X − II ( R ( X, Y ) Z, W ) − II ( Z, R ( X, Y ) W )Taking the tangent part yields ( ∇ X R )( Z, W ) Y = ( ∇ Y R )( Z, W ) X . Taking innerproduct with T ∈ T p M we have( ∇ R )( Z, W, Y, T, X ) = ( ∇ R )( Z, W, X, T, Y ) , that is, ∇ R is symmetric in the third and fifth entries. But ∇ R is also skew-symmetric in the third and fourth entries, so that ∇ R = 0. (cid:3) Lemma 15. Let ( M, g ) be a connected Riemannian manifold, and let Ω j : T M → V j , for j = 1 , be virtual immersions with skew-symmetric second fundamentalforms II j . Assume V , V are minimal in the sense that V j = span(Ω j ( T M )) . Then Ω , Ω are equivalent in the sense that there is a linear isometry L : V → V suchthat Ω = L ◦ Ω . IRTUAL IMMERSIONS AND MINIMAL HYPERSURFACES 11 Proof. Define a connection ˆ D on the vector bundle T M ⊕ ∧ T M byˆ D W (cid:16) Z, X i X i ∧ Y i (cid:17) = (cid:16) ∇ W Z − X i R ( X i , Y i ) W, W ∧ Z + ∇ W X i X i ∧ Y i (cid:17) Define bundle homomorphisms ˆΩ j : T M ⊕ ∧ T M → M × V j , for j = 1 , 2, byˆΩ j (cid:16) Z, X i X i ∧ Y i (cid:17) = (cid:16) p, Ω j ( Z ) + X i II j ( X i , Y i ) (cid:17) for Z, X i , Y i ∈ T p M . By Lemma 14(b), given vector fields X i , Y i , Z, W , we have(9) ( D j ) W (cid:16) ˆΩ i (cid:16) Z, X i X i ∧ Y i (cid:17)(cid:17) = ˆΩ j (cid:16) ˆ D W (cid:16) Z, X i X i ∧ Y i (cid:17)(cid:17) where D j denotes the natural flat connection on M × V j . This implies that theimage of ˆΩ j is D j -parallel, and hence, by minimality of V j , that ˆΩ j is onto M × V j .Define a bundle isomorphism L : M × V → M × V by L (cid:16) ˆΩ (cid:16) Z, X i X i ∧ Y i (cid:17)(cid:17) = ˆΩ (cid:16) Z, X i X i ∧ Y i (cid:17) for Z, X i , Y i ∈ T p M . This is well-defined because, by Lemma 14(a), ker ˆΩ = ker ˆΩ .Indeed, they are both equal to n(cid:16) , X i X i ∧ Y i (cid:17) (cid:12)(cid:12)(cid:12) X i , Y i ∈ T p M, X a,b R ( X a , Y a , X b , Y b ) = 0 o We claim the linear map L p : V → V is independent of p ∈ M . Indeed, giventwo points p, q ∈ M , choose a curve γ ( t ) in M joining p to q . Choose smooth vectorfields Z, X i , Y i along γ ( t ) such that ˆΩ ( Z, P X i ∧ Y i ) is constant equal to v ∈ V .Then, by (9), ˆ D ˙ γ ( Z, P X i ∧ Y i ) ⊂ ker ˆΩ . But by Lemma 14(a), ker ˆΩ = ker ˆΩ .Therefore, again by (9), we see that L ( v ) is constant along γ , so that L p = L q .Calling this one linear map L , we have ˆΩ = L ◦ ˆΩ by construction. In particular,Ω = L ◦ Ω , finishing the proof that Ω and Ω are equivalent. (cid:3) Proof of Theorem B. Let ( M, g ) be a compact Riemannian manifold. If M is sym-metric, then it admits a virtual immersion with skew-symmetric second fundamen-tal form by Lemma 11. Conversely, let Ω : T M → V be a virtual immersion withskew-symmetric second fundamental form. We may assume that V is minimal, andthen uniqueness of Ω follows from Lemma 15. What remains to be proved is that M is symmetric.By Lemma 14(c), M is locally symmetric, and therefore its universal cover ˜ M is symmetric. By Lemma 14(a), ˜ M has non-negative curvature, so that ˜ M splitsisometrically as ˜ M = N × R l , where N is a compact, simply-connected symmetricspace.Denoting by G the isometry group of N , we claim that Isom( ˜ M ) = G × Isom( R l ).Indeed, tangent vectors of the form ( v, 0) are characterized by the fact that theassociated geodesic has bounded image. Thus, any isometry γ of N × R l preservesthe splitting T ( N × R l ) = T N ⊕ R l . In particular, fixing p ∈ N , the maps g : N → N and B : R l → R l given by composing the obvious maps g : N → N × { } ֒ → N × R l γ → N × R l → NB : R l → { p } × R l ֒ → N × R l γ → N × R l → R l are isometries. Since γ and g × B are isometries of N × R l whose values and firstderivatives coincide at ( p, γ = g × B .Denote by Γ ⊂ Isom( ˜ M ) = G × Isom( R l ) the group of deck transformations forthe covering ρ : ˜ M → M (so that Γ is isomorphic to π ( M )). Let ( p, ξ ) ∈ N × R l ,and consider the symmetry s = s p × s ξ at ( p, ξ ). We need to show that s normalizesΓ, so that s descends to a well-defined symmetry of M . We will in fact show that sγs = γ − for every γ = g × B ∈ Γ.Note that Ω × Id : T ˜ M → g × R l is a virtual immersion with skew-symmetricsecond fundamental form, where Ω is defined as in (8). Since the pull-back ρ ∗ Ωis again such a virtual immersion, Lemma 15 implies that Ω × Id is fixed by Γ.Therefore, B must be a translation. As for g ∈ G , we have Ad g X = X for every X ∈ m , and, since [ m , m ] = h , the same equation holds for every X ∈ g . Inparticular, g commutes with the identity component G of G .Since G acts transitively on N , this implies that the displacement function q ∈ N d ( q, g ( q )) has a constant value d (in other words, it is a Clifford-Wolftranslation). Moreover, the isometries s p gs p and gs p gs p also commute with G ,so that they are Clifford-Wolf translations as well. Thus it suffices to show that gs p gs p has a fixed point, for this would imply that gs p gs p = Id, and hence that γsγs = Id.Let c ( t ) : [0 , → N be a minimal geodesic between c (0) = p and c (1) = g ( p ), andlet m = c (1 / 2) denote the midpoint. Then, the concatenation of c with g − c mustbe a geodesic, because d ( g − m, m ) = d = d ( g − m, p )+ d ( p, m ). Thus, extending thegeodesic segment c to a complete geodesic c : R → N , we see that c ( − 1) = g − ( p ).In particular, s p gs p ( p ) = s p g ( p ) = s p ( c (1)) = c ( − 1) = g − ( p ), and hence gs p gs p fixes the point p ∈ N , finishing the proof. (cid:3) Affine bounds on the index Let Σ → M = G/H be a compact, immersed, minimal hypersurface in a compactsymmetric space. This section addresses the question of when the linear bound in b (Σ) on the extended index of Σ given in Theorem A can be “improved” to anaffine bound on the index. To find such affine bounds, we consider the uniquevirtual immersion Ω : T M → g with skew-symmetric second fundamental form,defined in (8). In view of Proposition 10, it suffices to find an upper bound on thedimension of the space of harmonic 1-forms ω on Σ such that X ij ( ω ) lies in thekernel of the Jacobi operator J Σ for all i, j , where X ij is defined in (6).To compute J Σ ( X ij ( ω )) at p ∈ Σ, choose an orthonormal frame E , . . . E n − ofΣ, such that ( ∇ Σ E i E j ) p = 0, and an orthonormal basis θ i of g . Then J Σ ( X ij ( ω )) = ∇ ⊥ E k ∇ ⊥ E k (cid:0) h N ∧ ω , θ i ∧ θ j i N (cid:1) + ( | A | + Ric M ( N, N )) h N ∧ ω , θ i ∧ θ j i N = (cid:10) D E k D E k ( N ∧ ω ) + ( | A | + Ric M ( N, N ))( N ∧ ω ) , θ i ∧ θ j (cid:11) N. Here and in the rest of the section we adopt the convention that, when repeated in-dices appear, we are summing over them. From the previous equation J Σ ( X ij ( ω )) =0 for all i, j if and only if(10) D E k D E k ( N ∧ ω ) + ( | A | + Ric M ( N, N ))( N ∧ ω ) = 0 . IRTUAL IMMERSIONS AND MINIMAL HYPERSURFACES 13 We proceed now to compute Equation (10). For this, let ∇ denote the LeviCivita connection of Σ. We have: D E k D E k ( N ∧ ω ) = D E k (cid:16) − S N E k ∧ ω + II ( E k , N ) ∧ ω (11) + N ∧ ∇ E k ω + N ∧ II ( E k , ω ) (cid:17) We compute the derivatives of each of the four summands on the right hand sideof the equation above, in (a)–(d) below. For the sake of clarity, when computingterms of the type D E k ( X ∧ Y ), we display the result in the form (( D E k X ) ∧ Y ) +( X ∧ ( D E k Y )), that is, we write the two parts separately inside parentheses. D E k ( − S N E k ∧ ω ) = (cid:16) − ∇ E k ( S N E k ) ∧ ω − | S N | N ∧ ω (a) − II ( E k , S N E k ) ∧ ω (cid:17) + (cid:16) − S N E k ∧ ∇ E k ω − S N E k ∧ (cid:10) S N ω , E k (cid:11) N − S N E k ∧ II ( E k , ω ) (cid:17) D E k ( II ( E k , N ) ∧ ω ) = (cid:16) − R ( E k , N ) E k ∧ ω − II ( E k , S N E k ) ∧ ω (cid:17) (b) + (cid:16) II ( E k , N ) ∧∇ E k ω + II ( E k , N ) ∧ (cid:10) S N E k , ω (cid:11) N + II ( E k , N ) ∧ II ( E k , ω ) (cid:17) In the equation above, it was used the fact that ( D Z II )( X, Y ) = − R ( X, Y ) Z , andthat by assumption ∇ E k E j = 0 at p . D E k ( N ∧ ∇ E k ω ) = (cid:16) − S N E k ∧ ∇ E k ω + II ( E k , N ) ∧ ∇ E k ω (cid:17) (c) + (cid:16) N ∧ ∇ E k ∇ E k ω + N ∧ II ( E k , ∇ E k ω ) (cid:17) D E k ( N ∧ II ( E k , ω )) = (cid:16) − S N E k ∧ II ( E k , ω ) + II ( E k , N ) ∧ II ( E k , ω ) (cid:17) (d) + (cid:16) − N ∧ R ( E k , ω ) E k + N ∧ II ( E k , ∇ E k ω )+ N ∧ II ( E k , (cid:10) S N ω , E k (cid:11) N ) (cid:17) Lemma 16. Let M = G/H be a compact symmetric space and let Σ → M aclosed, immersed, minimal hypersurface. Suppose ω is a harmonic one-form on Σ such that J Σ ( X ij ( ω )) = 0 for all ≤ i < j ≤ d . Thena) The operators ∇ ω and S N commute.b) At any point p = J g K ∈ G/H , Ad − g II ( ω , N ) is contained in the center of h , z ( h ) = { x ∈ h | [ x, y ] = 0 ∀ y ∈ h } .c) For any vector x tangent to Σ , ∇ x (cid:0) II ( ω , N ) (cid:1) = − R ( ω , N ) x . In particular, k II ( ω , N ) k is constant. For example, when the symmetric space M is a torus, parts (b) and (c) aretrivially satisfied, and part (a) is equivalent to Proposition 3 in [ACS17]. Remark . In fact, J Σ ( X ij ( ω )) = 0 for all 1 ≤ i < j ≤ d if and only if conditions(a), (b), (c) are satisfied. We omit the proof of the reverse implication since it isnot used in the remainder of this article. Proof. Notice that Equation (10) is a vector-valued equation in ∧ V . Fixing a point p in Σ and identifying T p M with its image under Ω p in V , we can split orthogonally V = T p M ⊕ ν p M = R · N ⊕ T p Σ ⊕ ν p M and this induces a splitting of ∧ V . Thedifferent parts of the lemma follow from projecting Equation (10) on the differentsubspaces.a) Projecting Equation (10) onto the subspace ∧ T p Σ ⊂ ∧ V and using thecomputations above, we obtain0 = − ∇ E k ( S N E k ) ∧ ω − S N E k ∧ ∇ E k ω − π Σ ( R ( E k , N ) E k ) ∧ ω . (12)Here π Σ denotes orthogonal projection onto T p Σ. Applying Codazzi equation tothe first term we get ∇ E k ( S N E k ) = h∇ E k ( S N E j ) , E k i E j = (cid:0)(cid:10) ∇ E j ( S N E k ) , E k (cid:11) + h R ( E j , E k ) E k , N i (cid:1) E j = ( E j h S N E k , E k i − h R ( E k , N ) E k , E j i ) E j = − π Σ ( R ( E k , N ) E k )The last equation holds because, since Σ is minimal, the first summand in thesecond equation vanishes. Equation (12) thus becomes0 = π Σ ( R ( E k , N ) E k ) ∧ ω − S N E k ∧ ∇ E k ω − π Σ ( R ( E k , N ) E k ) ∧ ω ⇒ S N E k ∧ ∇ E k ω For any x, y ∈ T p Σ, we thus have0 = (cid:10) S N E k ∧ ∇ E k ω , x ∧ y (cid:11) = h S N E k , x i (cid:10) ∇ E k ω , y (cid:11) − h S N E k , y i (cid:10) ∇ E k ω , x (cid:11) Clearly S N is symmetric. Since ω is harmonic, ∇ ω is symmetric as well, and theequation above becomes0 = h E k , S N x i (cid:10) ∇ y ω , E k (cid:11) − h E k , S N y i (cid:10) ∇ x ω , E k (cid:11) = (cid:10) S N x, ( ∇ ω ) y (cid:11) − (cid:10) S N y, ( ∇ ω ) x (cid:11) = (cid:10) [ S N , ∇ ω ] x, y (cid:11) Since this holds for every x and y , the result follows.b) Projecting Equation (10) onto the subspace ∧ ν p M ⊂ ∧ V one gets II ( N, E k ) ∧ II ( ω E k ) = 0 . (13)Taking inner product with elements II ( x, y ) ∧ II ( u, v ) (such elements span ∧ ν p M ),one gets0 = h II ( x, y ) , II ( N, E k ) i (cid:10) II ( u, v ) , II ( ω , E k ) (cid:11) − (cid:10) II ( x, y ) , II ( ω , E k ) (cid:11) h II ( u, v ) , II ( N, E k ) i = h R ( x, y ) N, E k i (cid:10) R ( u, v ) ω , E k (cid:11) − (cid:10) R ( x, y ) ω , E k (cid:11) h R ( u, v ) N, E k i = (cid:10) R ( x, y ) N, R ( u, v ) ω (cid:11) − (cid:10) R ( x, y ) ω , R ( u, v ) N (cid:11) IRTUAL IMMERSIONS AND MINIMAL HYPERSURFACES 15 It is easy to check that, given η = II ( x, y ), then S η = R ( x, y ). The equation abovethen becomes 0 = (cid:10) [ S η , S η ] N, ω (cid:11) , η = II ( x, y ) , η = II ( u, v )Since Ω( T M ) = g , equation above holds for any η , η normal vectors. Using Ricciequation, this implies that (cid:10) R ⊥ ( ω , N ) η , η (cid:11) = 0and in particular R ⊥ ( ω , N ) η = 0 for all η in ν p M . From Lemma 11, letting p = J g K , then ν p M = Ad g h and, letting η = Ad g v for v ∈ h , one has[ II ( ω , N ) , η ] = 0 ∀ η ∈ Ad g h ⇒ [ Ad − g II ( ω , N ) , v ] = 0 ∀ v ∈ h Therefore Ad − g II ( ω , N ) belongs to the center of h .c) Projecting Equation (10) onto the subspace T p Σ ⊗ ν p M ⊂ ∧ V we get0 = − II ( E k , S N E k ) ∧ ω + S N E k ∧ II ( E k , ω ) − II ( E k , N ) ∧ ∇ E k ω ) . (14)The first term can be rewritten as II ( E k , E j ) h S N E k , E j i ∧ ω . However, the leftfactor of the exterior product is skew symmetric in i, k and therefore the ( j, k )-termin the sum cancels with the ( k, j )-term. This term thus vanishes, and Equation (14)is equivalent to S N E k ∧ II ( E k , ω ) + ∇ E k ω ∧ II ( E k , N ) = 0 . Taking the inner product with an element of the form x ∧ II ( y, z ) for x ∈ T p Σ and y, z ∈ T p M (these elements span the whole of T p Σ ⊗ ν p M ) one gets0 = (cid:10) S N E k ∧ II ( E k , ω ) + ∇ E k ω ∧ II ( E k , N ) , x ∧ II ( y, z ) (cid:11) = h S N E k , x i (cid:10) II ( E k , ω ) , II ( y, z ) (cid:11) + (cid:10) ∇ E k ω , x (cid:11) h II ( E k , N ) , II ( y, z ) i = − h S N x, E k i (cid:10) R ( y, z ) ω , E k (cid:11) − (cid:10) ∇ x ω , E k (cid:11) h R ( y, z ) N, E k i = − (cid:10) S N x, R ( y, z ) ω (cid:11) − (cid:10) ∇ x ω , R ( y, z ) N (cid:11) = − (cid:10) R ( ω , S N x ) y, z (cid:11) + (cid:10) R ( ∇ x ω , N ) y, z (cid:11) = (cid:10) R ( ω , ∇ x N ) y + R ( ∇ x ω , N ) y, z (cid:11) = (cid:10) II ( ω , ∇ x N ) + II ( ∇ x ω , N ) , II ( y, z ) (cid:11) Therefore it follows that0 = II ( ω , ∇ x N ) + II ( ∇ x ω , N ) = ∇ x ( II ( ω, N )) − R ( ω , N ) x. (cid:3) Theorem 18. Let M = G/H be a compact symmetric space, Σ → M a closed,immersed, minimal hypersurface, and let H denote the space of harmonic one-forms ω on Σ satisfying the following two conditions at all points p ∈ Σ .a) The operators ∇ ω and S N commute.b) R ( ω , N, ω , N ) = 0 .Then ind(Σ) ≥ (cid:18) d (cid:19) − (cid:0) b (Σ) − dim H − dim z ( h ) (cid:1) where z ( h ) denotes the center z ( h ) = { x ∈ h | [ x, y ] = 0 ∀ y ∈ h } . Proof. Let H be the space of harmonic one-forms on Σ that satisfy the threeconditions listed in Lemma 16. Then H is a subspace of H . More precisely,let p = [ e ] ∈ M = G/H , and define the linear map Z : H → z ( h ) by Z ( ω ) = II p ( ω , N ). By condition (c) in Lemma 16, H equals the kernel of Z . Thereforeits codimension is at most dim z ( h ), and the result follows from Lemma 16 andProposition 10. (cid:3) Corollary 19. Let M n = G/H , n > , be a CROSS, and Σ → M a compact,immersed, minimal hypersurface. Then ind(Σ) ≥ (cid:18) dim G (cid:19) − b (Σ) . Proof. The CROSSes M = G/H are S n = SO( n + 1) / SO( n ), RP n = O( n +1) / O( n ), CP n = SU( n +1) / SU( n ), HP n = Sp( n +1) / Sp( n ) and Ca P = F / Spin(9).In all these cases, h is semisimple and hence centerless. Now the result follows fromthe fact that M has positive sectional curvature, together with Theorem 18. (cid:3) Remark . The only 2-dimensional CROSSes are S and RP . In these cases, aminimal hypersurface Σ is a closed geodesic, so that b (Σ) = 1. Since the index isnon-negative, in particular one has ind(Σ) ≥ b (Σ) − Proof of Theorem C(b). Define δ i for i = 1 , δ i = 1 if dim M i = 2, and δ i = 0otherwise. Note that D = 1 + δ + δ , and that δ + δ is the dimension of thecenter of h .Suppose first that Σ is of the form M × Σ , where Σ is a minimal, compacthypersurface in M . By Corollary 19 and Remark 20, the index of Σ is boundedbelow by (cid:0) dim G (cid:1) − ( b (Σ) − δ ), In this case we have b (Σ) = b (Σ ), andind(Σ) ≥ ind(Σ ) ≥ (cid:18) dim G (cid:19) − ( b (Σ ) − δ ) ≥ (cid:18) dim G × G (cid:19) − ( b (Σ) − − δ − δ )thus the proposition is proved in this case. Clearly the same argument would workin the case Σ = Σ × M , where Σ is a minimal compact immersed hypersurfacein M .Suppose now that Σ is not as before. Then, by Theorem 18, it suffices to showthat the space of harmonic one-forms ω on Σ such that R ( ω , N, ω , N ) = 0 is atmost one-dimensional.Let ω , ω be two such harmonic one-forms. Since Σ is neither of the formΣ × M nor of the form M × Σ , there exists an open set U ⊂ Σ such that forevery p = ( p , p ) ∈ U , the normal vector N p ∈ T p M ⊕ T p M is not tangentto neither T p M nor T p M . In particular, for every p ∈ U there exists a uniquezero curvature plane π p through N p , and in particular a unique direction in π p ,perpendicular to N p . It thus follows that ω and ω are collinear in U : ω = f ω for some function f : U → R . However, since ω , ω are both closed and co-closed,it is easy to check that df must be both parallel and normal to ω in U , and inparticular df = 0. Thus f is constant, and ω , ω are linearly dependent in U ,hence linearly dependent on the whole of Σ, a contradiction. (cid:3) IRTUAL IMMERSIONS AND MINIMAL HYPERSURFACES 17 Proof of Theorem C(a). By Theorem 18, it suffices to show that dim H ≤ r − ω ∈ H . The fact that ∇ ω and S commute is equivalent to condition (2.3) in[ACS17, Proposition 5], and from that proposition it follows that ω is determinedby its value and the value of its covariant derivative at any point p .Let p ∈ Σ where all the principal curvatures are distinct. We may assume that N ( p ) ∈ T p M belongs to a regular (that is, principal or exceptional) orbit underthe isotropy representation of G p on T p M . Indeed, the set of singular vectors hascodimension at least two, and, since the shape operator has distinct eigenvalues,its image has codimension at most one. This means that if N ( p ) is singular, thenthere is a nearby p ′ ∈ Σ such that N ( p ′ ) is regular.Define V = { X ∈ T Σ | II ( X, N ) = 0 } = { X ∈ T Σ | R ( X, N, X, N ) = 0 } . Since N ( p ) is regular, V is a smooth distribution of rank r − p . Moreover, for any ω ∈ H , we have ω ∈ V .Let { e , . . . , e n } be an orthonormal frame of eigenvectors for the shape operator,defined on an open neighbourhood of p , with S ( e i ) = a i e i . Since ∇ ω commuteswith S , there are functions λ i defined near p , such that ∇ e i ω = λ i e i . Differenti-ating the equation II ( ω , N ) = 0 and using Lemma 14, we obtain λ i II ( e i , N ) = − a i II ( ω , e i ) . 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