The Morse Index for Manifolds with Constant Sectional Curvature

. We compute the Morse index of a critical submanifold of the energy functional on the loop space of a manifold with constant sectional curvature. The case of constant non-positive sectional curvature is a known result and the case of a sphere has been proved by Klingenberg. We adapt Klingenberg’s proof of the case of a sphere to the case of constant sectional curvature, to obtain the possible Morse indices of critical submanifolds of the energy functional.


Introduction
The Morse index of a critical submanifold of the energy functional on the loop space of a Riemannian manifold has been computed in the literature.It has been computed by Klingenberg for spheres in [7] and for complex projective spaces, quaternionic projective spaces, and Cayley projective planes in [6].The indices for compact rank one symmetric spaces (which are spheres, real projective spaces, complex projective spaces, quaternionic projective spaces, and the Cayley projective plane) have been computed by Ziller [16].The indices for simply connected compact rank one symmetric spaces have also been computed by Hingston [3]. 1he Morse indices of the critical submanifolds of the energy functional have been used in the literature to obtain obstruction results for the embeddings of Lagrangian tori (see [13]), spheres and other compact rank one symmetric spaces (see [11]) utilizing an index relation (see [1,5,15]) involving the Morse index.They also are used to obtain restrictions on the Maslov class of Lagrangian submanifolds [4].The result obtained in this paper modifies the possible Morse indices listed in [12] where the use of Morse indices provided an alternative proof of a nonexistence result for certain displaceable constant sectional curvature Lagrangian submanifolds in symplectically aspherical symplectic manifolds (M 2n , ω) with n > 4 [12]. 2n this paper, we adapt the proof of [7] for the case of spheres to manifolds with constant sectional curvature to obtain Theorem 1.1.In Theorem 1.1, the case A ≤ 0 is a classical result, a consequence of the known more general statement that the Morse index I is zero if M n is a manifold of nonpositive sectional curvature.From the remaining cases, the ones not overlapping with Klingenberg's, Hingston's and Ziller's results are new to the best of my knowledge.
2 dt where q(t) ∈ Λ(M n ), then the Morse index I of the nondegenerate critical submanifold S c (containing c) of the energy functional is as follows: where

Preliminaries
The loop space Λ(M n ) is defined as the set of H 1 -maps of S 1 into M n (see [3,7,16]).
The following definitions agree with their corresponding ones in [7]: and ∇ξ is the covariant derivative ∇ ċξ.
T c Λ(M n ) has an orthogonal decomposition into subspaces spanned by the eigenvectors of A c having positive, zero and negative eigenvalue, respectively.The dimension of Define the map χ : If the index of c is the same constant for all c in a critical submanifold S, this constant is called the (Morse) index of S. If the nullity of D 2 E g (c) is constant for all c ∈ S, then this number is called the nullity of S.
S is called a nondegenerate critical submanif old if the critical submanifold S has an index and if the nullity of S is equal to the dimension of S.

Known Results
Klingenberg computed the index of a critical submanifold of the energy functional on the loop space of the sphere as given in the following theorem: 2 dt on the loop space Λ(S n ) decomposes into the nondegenerate critical submanifolds Λ 0 S n which is isomorphic to S n and B q consisting of the q-fold covered great circles q = 1, 2, .... B q is isomorphic to the Stiefel manifold V (2, n + 1) of orthonormal 2-frames in R n+1 and the index of B q is (2q − 1)(n − 1).
As stated in the introduction, the index calculated for the sphere case in Theorem 3.1 has also been calculated for the complex projective space, the quaternionic projective space and the Cayley projective plane by Klingenberg in [6] and by Hingston in [3].This is given in Theorem 3.2.The indices for compact rank one symmetric spaces are given by Ziller in [16].Theorem 3.2 (p.103 of [2], p.19 of [5]).The standard metric on M is normalized so that the maximal sectional curvature is 1.Then the critical set of the energy function on ΛM consists of the critical manifolds A m of geodesics of length 2πm, m ≥ 0. We have A 0 ∼ = M and A m ∼ = ST M the unit tangent bundle of M for m ≥ 1.By counting zeros of the Jacobi fields, one can see that A m (m > 0) is a nondegenerate critical submanifold of index where a = n, 2, 4, or 8, respectively, for M = S n , CP n , HP n , CaP 2 and s = dimM a (= 1, n, n, n = 2, respectively).
In addition, for M = RP n , the index of a nondegenerate critical submanifold A m of m-fold covered geodesics for m ≥ 1 is (m − 1)(n − 1), by Ziller [16].

Proof of Theorem 1.1
Let c(t) ∈ Λ(M n ) be a critical point of E g .For a Riemannian manifold with constant sectional curvature A, the formula of the Riemann curvature endomorphism for X = η(t) in the tangent space [8].To find the Morse index, we will count the periodic solutions of for λ < 0 [7]. 3ere, we use the notation Rη = R(η(t), ċ(t), ċ(t)) 4 and ∇ is the covariant derivative on tangent bundle T M, derived from the Levi-Civita connection [7].
When A = 0, we have R = 0, and Eq.(4.2) reduces to Since λ = 1 is not an eigenvalue, the equation can be written as This equation has no periodic solutions when λ < 0 (p.59 of [7]); so the dimension of the subspace of the tangent space T c ΛM spanned by the eigenvectors having negative eigenvalue, dim T − c ΛM , for c in a nondegenerate critical submanifold is zero.
Then, the last formula decomposes into (ver) Solutions of (tan) for λ < 0: • If λ < 0, then there are no periodic solutions.Solutions of (ver) for λ < 0: then there are no periodic solutions. 5 • If −A ċ 2 ≤ λ < 0, then we consider Eq. (4.6) on the universal cover of M. The universal cover of a constant sectional curvature manifold with positive A is S n by the Killing-Hopf theorem.The general solution of (4.6) considered on the universal cover is of the form 67 Here, if c denotes a lift of c (with ċ = ċ ) and Ω M denotes the space of paths on the universal cover M , as it is the case with ξ, ξ 0 and ξ 1 also belong to T cΩ M and are vector fields along c which are orthogonal to ċ and are induced by parallel transport.

Counting the periodic solutions to find dimT −
c ΛM For λ < 0: Case 1: When −A ċ 2 ≤ λ < 0 is not satisfied, then there are no periodic solutions.(This will always be the case for A < 0 and may be possible for A > 0.) 5 Equation (4.6) is of the form ∇ 2 ξ + Eξ = 0 where E = A ċ 2 +λ 1−λ .If λ < −A ċ 2 , then E < 0. Let ċ, X 1 , ..., X n−1 be an orthonormal frame invariant by parallelism along a geodesic c.If E < 0, Y i = sinh( √ −Es)X i 's and Z i = cosh( √ −Es)X i 's form a basis for the solutions of ∇ 2 ξ + Eξ = 0 that are orthogonal to ċ. (s denotes the arclength).(see p.485-486 of [2].)Noting that these solutions are not periodic, we conclude that there are no periodic solutions orthogonal to ċ. 6 With the notation in Footnote 5, if −A ċ 2 ≤ λ < 0, then E ≥ 0.If E > 0, Y i = sin( √ Es)X i 's and Z i = cos( √ Es)X i 's form a basis for the solutions of ∇ 2 ξ + Eξ = 0 that are orthogonal to ċ [2].If E = 0, then X i 's form a basis for the solutions [2]. 7For S n with A = 1, letting A = 1 in Eq. 4.8, we obtain the same ξ(t) obtained by Klingenberg in [7].