Fields Institute Communications Volume 00, 0000

Ordinary Differential Equations of Morse–Sturm Type.

F. Mercuri, P. Piccione, D. V. Tausk

Dedicated to Professor Waldyr Oliva for his 70th birthday

1 Introduction

Let M = M^{n} be a n-dimensional Riemannian manifold with metric tensor g, and p, q ∈ M. Let Ω(p, q) denote the space of H^{1} curves γ : [0, 1] → M, joining p and q. It is well known that the critical points of the energy functional:

E : Ω(p, q) → IR,  E(γ) = \frac{1}{2} ∫_{0}^{1} g(γ̇(t), γ̇(t)) dt  (1)

are the geodesics joining p and q. The tangent space at a curve γ is the space T_{γ}Ω(p, q) of H^{1} vector fields along γ vanishing at the endpoints, and the Hessian of the functional E at a critical point γ is given by the symmetric bilinear form:

I(V, W) = ∫_{0}^{1} [g(V′, W′) + g(R(γ̇, V)γ̇, W)] dt, V, W ∈ T_{γ}Ω(p, q), (2)

where R denotes the curvature tensor of the Levi–Civita connection and primes denote covariant derivative along γ; here, we choose the sign convention

R(X, Y) = [∇_{X}, ∇_{Y}] − ∇_{[X,Y]}.

Integrating by parts we have, on the subspace of H^{2} vector fields vanishing at the endpoints:

I(V, W) = ∫_{0}^{1} g(J (V), W ) dt, J (V) = −V′′ + R(γ̇, V)γ̇;

the differential operator J is known as the Jacobi operator. Vector fields along γ that are solutions of the Jacobi equation J (V) = 0, are called Jacobi fields along γ.

In the context of Riemannian geometry, the geodesic variational problem just described is well understood; when passing to the case of semi-Riemannian geodesics a series of new phenomena appear, and the investigation of some of them is the main subject of this paper. Recall that a semi-Riemannian (also called pseudo-Riemannian) manifold is a differentiable manifold endowed with a nondegenerate (possibly non positive definite) smooth metric tensor g. Semi-Riemannian geodesics are critical points of the geodesic energy functional (1) as in the Riemannian case.

\begin{matrix}c \\ ◯ \end{matrix}0000 American Mathematical Society

Research partially supported by CNPq and FAPESP, Brazil.

However, when the metric g is not positive definite, the functional E is no longer bounded from below, the index of the index form (2) is always infinite, and the Jacobi differential operator J is not in general self-adjoint with respect to a (positivedefined) Hilbert space inner product on T_{γ}Ω(p, q).

Using a parallel frame along a geodesic γ, we identify the Jacobi equation J (V) = 0 with the following second-order linear homogeneous ODE in IR^{n}:

J (J) = −J′′(t) + R(t)J(t) = 0.  (3)

In equation (3), J is an IR^{n}-valued map on [0, 1] and R : [0, 1] → L(IR^{n}, IR^{n}) is a smooth map with R(t) symmetric with respect to a fixed non degenerate symmetric bilinear form ⟨·, ·⟩ on IR^{n}; the primes in (3) denote standard derivatives. Equation (3) is called a Morse–Sturm system.

We will consider the symmetric bilinear form:

I(V, W) = ∫_{0}^{1} [⟨V′(t), W′(t)⟩ + ⟨R(t)V (t), W (t)⟩] dt,  (4)

associated to the ODE (3). The bilinear form (4) is defined in the Hilbert space H^{1}([0, 1], IR^{n}) of IR^{n}-valued maps on the interval [0, 1] of class H^{1}.

We remark that every system of the type (3) arises (by means of a parallel frame) from the Jacobi equation along a geodesic in a suitable semi-Riemannian manifold (see [9, 10]).

For t ∈ (0, 1] we set:

𝕁(t) = {J(t) : J (J) ≡ 0, J(0) = 0}.  (5)

The instant t ∈ (0, 1] is called conjugate, if m(t) = dim(𝕁(t)^{⊥}) > 0. The integer m(t) is called the multiplicity of the conjugate instant t.

We recall that the index of a symmetric bilinear form B : V × V → IR on a real vector space V is defined by:

n_{−}(B) = sup {dim(W) : W is a subspace of V on which B is negative definite};

the coindex of B is defined by n_{+}(B) = n_{−}(−B), and the signature of B is defined by sgn(B) = n_{+}(B) − n_{−}(B). We now define:

• the Morse index of γ (or of a Morse–Sturm system (3)) by:

i_{Morse} = i_{Morse}(γ) = n_{−}(I);

• the geometric index of γ (or of a Morse–Sturm system (3)) by:

i_{geom} = i_{geom}(γ) = ∑_{t∈(0,1)} m(t).

The classical Theorem of M. Morse (see, for instance, [3]) can be stated as follows:

Theorem 1.1 (Morse) If γ : [0, 1] → M is a geodesic on a Riemannian manifold (M, g) then the conjugate points along γ are isolated and

i_{Morse} = i_{geom} < ∞.

The theorem above can also be stated for Morse–Sturm systems (3) for which the coefficient R is symmetric with respect to a positive-definite inner product ⟨·, ·⟩ in IR^{n}.

In the case where ⟨·, ·⟩ is positive-definite (which we shall refer to as the Riemannian case), the Jacobi operator J, densely defined in H = L^{2}([0, 1], IR^{n}), is a self-adjoint operator which defines a spectral decomposition:

H = ⨁_{λ∈σ(J)} H_{λ},

where H_{λ} are finite dimensional eigenspaces of J and the spectrum σ(J) of J is discrete and bounded from below. The index i_{Morse} of the bilinear form I is the sum of the dimensions of the eigenspaces H_{λ} relative to the negative eigenvalues λ of J.

Another interesting aspect of the Riemannian case is that the form I in the Hilbert space H^{1}([0, 1], IR^{n}) is the bilinear form associated to a self-adjoint operator of the form Id + K with K compact. From the spectral theorem for self-adjoint compact operators it follows that if K_{n} is a sequence of self-adjoint compact operators converging to K and if I_{n} is the symmetric bilinear form corresponding to Id + K_{n}, then:

n_{−}(I_{n}) = n_{−}(I),

for n sufficiently large, provided that Id + K is invertible. In particular we get:

Theorem 1.2 If γ_{n} : [0, 1] → M is a sequence of geodesics converging to a geodesic γ : [0, 1] → M on a Riemannian manifold (M, g) and if t = 1 is not conjugate along γ, then i_{Morse}(γ) = lim_{n→∞} i_{Morse}(γ_{n}).

One can also state a version of the theorem above for Morse–Sturm systems; such version states that (in the Riemannian case) if t = 1 is not conjugate for (3) then i_{Morse} is stable by uniformly small perturbations of the coefficient R of the system.

By the Morse Index Theorem, one obtains immediately the following stability result for the number of conjugate points along a geodesic:

Corollary 1.3 Under the assumptions of Theorem 1.2, we have:

i_{geom}(γ) = lim_{n→∞} i_{geom}(γ_{n}).

The aim of this paper is to discuss results like Theorems 1.1, 1.2 and Corollary 1.3 for semi-Riemannian geometry; equivalently, we work with Morse–Sturm systems (3) where the coefficient R is symmetric with respect to an arbitrary (possibly non positive definite) nondegenerate symmetric bilinear form ⟨·, ·⟩ on IR^{n}. The ideas are not completely new, as well as several results on this line, but the latter are, sometimes, stated imprecisely in the literature. In Section 2 we set the notation and state the extension of the Morse Index Theorem. Then, in Section 3 we describe the Maslov Index of the problem and give an idea of the proof of the index theorem. In Section 4 we discuss the stability properties of the index. Complete proofs are given in [10] which contains most of the material presented in this article. We give here a short list of references containing some recent results related to the theory discussed in this article. The notion of Maslov index in the context of semi-Riemannian geodesics was given in [9] although the appearance of the notion of Maslov index in the context of Lagrangian variational problems and Hamiltonian systems is much older (see for instance [1] and [5]). Some other extensions of the Morse Index theorem in semi-Riemannian geometry, as well as applications to the Morse theory for semi-Riemannian geodesics, were recently developed in a series of articles [8, 12, 11].

2 The Jacobi equation in semi-Riemannian geometry

We consider a symmetric non degenerate bilinear form ⟨·, ·⟩ on IR^{n} and the equation:

−J′′(t) + R(t)J(t) = 0,  (6)

where the operators R(t) are symmetric with respect to ⟨·, ·⟩. As discussed in the introduction, a system of the type above is called a Morse–Sturm system and it arises from Jacobi equations along semi-Riemannian geodesics by means of parallel frames. Below we make a list of phenomena that may occur in the case where ⟨·, ·⟩ is not positive definite:

• conjugate points are not in general isolated. In fact it is shown in [13] that given a closed subset F ⊂ IR contained in (0, 1], there exist a spacelike geodesic γ : [0, 1] → M in a suitable Lorentzian manifold M such that F is the set of conjugate instants along γ.

• The index of the associated bilinear form I (see (4)) is always infinite if g is not positive definite. Namely, take any Y_{0} ∈ IR^{n} such that g(Y_{0}, Y_{0}) < 0. Now take any solution Y (t) of the Morse–Sturm system with Y (0) = Y_{0} and choose ε such that g(Y (t), Y (t)) < 0 for t ∈ [0, ε). Let f : [0, 1] → IR be a non zero smooth function supported in (0, ε). An easy calculation shows that:

I(fY, fY) = ∫_{0}^{1} f′(t)^{2}g(Y (t), Y (t)) dt < 0.

• The operator J is not, in general, symmetric with respect to any (positive definite) inner product in H, so it may have complex eigenvalues.

• It is, in principle, not at all obvious that we have a stability result as in Corollary 1.3.

For the first problem we have the following result:

Lemma 2.1 If t_{0} is a conjugate instant and ⟨·, ·⟩ is nondegenerate on 𝕁(t_{0}) (recall (5)), then t_{0} is an isolated conjugate instant.

The conjugate instants t for which ⟨·, ·⟩ is nondegenerate on 𝕁(t) are called nondegenerate conjugate instants.

Remark 2.2 We observe that if ⟨·, ·⟩ is positive definite then the nondegeneracy hypothesis in the statement of Lemma 2.1 is automatically satisfied. Also if R : [0, 1] → L(IR^{n}, IR^{n}) is analytic, the conjugate instants are isolated, being the zeros of a suitable non zero analytic function.

If the number of conjugate instants is finite, we define the geometric index of (6) by:

i_{geom} = ∑_{t∈(0,1)} sgn(⟨·, ·⟩|_{𝕁(t)^{⊥}}).

In the above formula, ⊥ denotes the orthogonal complement with respect to ⟨·, ·⟩. The integer number sgn(⟨·, ·⟩|_{𝕁(t)^{⊥}}) is called the signature of the conjugate instant t.

Remark 2.3 If ⟨·, ·⟩ is positive then sgn(⟨·, ·⟩|_{𝕁(t)^{⊥}}) = dim(𝕁(t)^{⊥}) = m(t). So the signature of a conjugate instant equals its multiplicity in the Riemannian case.

For the second and third problems of the list of phenomena, we consider again the space H = L^{2}([0, 1], IR^{n}) and the symmetric bilinear form:

g̃(V, W) = ∫_{0}^{1} ⟨V (t), W (t)⟩ dt.

Since ⟨·, ·⟩ is nondegenerate, so it is g̃. A real eigenvalue λ of J for which g̃ is nondegenerate on H_{λ} is called a nondegenerate eigenvalue of J.

Remark 2.4 It can be shown by transversality techniques (see [4]) that the nondegeneracy assumption on the eigenvalues of J is generic with respect to uniformly small perturbations of the coefficient R.

It is not difficult to prove the following (see [10]):

Lemma 2.5 The eigenvalues of the Jacobi operator J are isolated and their real part is bounded from below. Moreover, the eigenspaces of J are all finitedimensional.

This allows us to define the spectral index of the Morse–Sturm system (6):

i_{Morse} = ∑_{λ<0} sgn(g̃|_{H_{λ}}),

where H_{λ} is the eigenspace of J corresponding to the eigenvalue λ. Again we observe that if ⟨·, ·⟩ is positive definite, then sgn(g̃|_{H_{λ}}) = dim(H_{λ}); so, in the Riemannian case, i_{Morse} coincides with the Morse index of (6).

We can state now the following extension of the Morse Index Theorem:

Theorem 2.6 (Extended Morse Index Theorem) Suppose that all conjugate instants are nondegenerate and that all real negative eigenvalues of J are nondegenerate. Then, if t = 1 is non conjugate:

i_{Morse} = i_{geom}.

In Section 3 we will give a sketch of the proof of Theorem 2.6.

Remark 2.7 In essentially the same way, we can treat the focal problem, i.e. solutions of the Jacobi equation with initial conditions:

J(a) ∈ P, J′(a) + S(J(a)) ∈ P^{⊥},

where P is a subspace of IR^{n} and S : P → P is ⟨·, ·⟩-symmetric operator. This kind of initial conditions arises when one considers geodesics γ starting orthogonally to a given submanifold P; in this case, P corresponds to the tangent space T_{γ(0)}P and S corresponds to the second fundamental form of P at γ(0) in the normal direction γ′(0).

For this reason, the geometric index is sometimes called the focal index. Just observe that in this case the boundary conditions are in general not self-adjoint.

3 The Maslov Index

As it often happens, a good way to prove that two number are equal is to prove that both are equal to a third one. This will be our strategy.

Consider the fundamental matrix of the system (6):

ψ(t) : IR^{n} ⊕ IR^{n} → IR^{n} ⊕ IR^{n}, ψ(t)(J_{0}, J0^{′}) = (J(t), J′(t)),

where J(t) is the solution of (6) with initial data J(0) = J_{0}, J′(0) = J0^{′}. Also consider in IR^{n} ⊕ IR^{n}, the symplectic (i.e., antisymmetric nondegenerate bilinear) form:

ω((X_{1}, X_{2}), (Y_{1}, Y_{2})) = ⟨X_{1}, Y_{2}⟩ − ⟨X_{2}, Y_{1}⟩.

The subspace l_{0} = {0} ⊕ IR^{n} is a Lagrangian subspace i.e., dim(l_{0}) = n and ω|_{l_{0}} = 0. An easy calculation shows that ψ(t) preserves ω so that ψ(t)(l_{0}) is also Lagrangian for all t ∈ [0, 1].

Let Λ denote the set of Lagrangian subspaces of (IR^{2n}, ω); we call Λ the Lagrangian Grassmannian of the symplectic space (IR^{2n}, ω). Observe that ψ defines a curve:

l : [0, 1] → Λ, l(t) = ψ(t)(l_{0}).  (7)

Let Λ_{0} = {π ∈ Λ : π ∩ l_{0} ≠ {0}}. Then an instant t_{0} is conjugate if and only if l(t_{0}) ∈ Λ_{0}. Our aim is to study the intersections of the curve l with Λ_{0}. We start by recalling some basic facts about the Lagrangian Grassmannian of a symplectic space. Since all symplectic spaces are symplectomorphic, we consider only IR^{2n} endowed with its canonical symplectic structure, i.e., the imaginary part of the canonical Hermitian product of ℂ^{n}.

• The differentiable structure on Λ. Let π be a Lagrangian subspace and π^{⊥} a complementary Lagrangian subspace to π; using the symplectic form, one gets a canonical identification between π^{⊥} and the dual space π^{∗} of π. Any Lagrangian subspace which intersects trivially π^{⊥}, is the graph of a symmetric linear map π → π^{⊥} ≅ π^{∗} and conversely, the graph of such a map is a Lagrangian subspace which intersects trivially π^{⊥}. So the set of subspaces which intersect π^{⊥} trivially is naturally parameterized in an \frac{n(n+1)}{2} linear space. This construction gives a differentiable (analytic) atlas for Λ; the differential of any of these charts provides a canonical isomorphism between each tangent space T_{π}Λ and the space of symmetric bilinear forms on π.

• The subvariety Λ_{0}. The subset Λ_{0} is a stratified set Λ_{0} = ⋃_{k=1}^{n} Λ_{k}, where:

Λ_{k} = {π ∈ Λ : dim(π ∩ l_{0}) = k}.

The set Λ_{0} is a compact analytic subvariety containing an open dense part Λ_{1}, which is actually a smooth submanifold of Λ.

• The homogeneous structure on Λ. Any unitary transformation of ℂ^{n} maps the canonical basis of IR^{n} ⊕ {0} to a (complex) orthonormal basis of ℂ^{n} which is orthonormal with respect to the Hermitian product. Thus it determines a real n-dimensional subspace of ℂ^{n} which is Lagrangian, since the imaginary part of the Hermitian product vanishes on the basis. Conversely, any Lagrangian subspace is obtained in this way. So the unitary group U(n) acts transitively on the space of Lagrangian subspaces. Moreover the unitary transformations which fix IR^{n} ⊕ {0} are the ones which send the canonical basis of IR^{n} ⊕ {0} into another orthonormal basis of IR^{n} ⊕ {0}, i.e., the transformations corresponding to matrices with real entries. So the isotropy subgroup is the orthogonal group O(n), thought as the subgroup of U(n) consisting of matrices with real entries. Hence Λ may be identified with the quotient U(n)/O(n).

• The fundamental group of Λ. The map det^{2} : U(n)/O(n) → S^{1} is well defined and it is known from algebraic topology that it induces an isomorphism between the fundamental groups. So π_{1}(Λ) ≅ ℤ, and, by Hurewicz theorem, H_{1}(Λ; ℤ) ≅ ℤ. Moreover (Λ \ Λ_{0}) is homeomorphic to a vector space (see the first item) so the inclusion (Λ, ∅) → (Λ, Λ\Λ_{0}) induces an isomorphism in homology (in positive dimensions), by the long exact sequence of the pair. For the basic facts of algebraic topology we refer to [2].

At this point we define the Maslov index of the Morse–Sturm system (6) as follows: In (7) we have associated to (6) a curve l on Λ. Since there are no conjugate instants near 0, for ε sufficiently small, l( (0, ε] ) is contained in Λ \ Λ_{0}. Also, if 1 is not a conjugate instant, then l(1) ∈ (Λ \ Λ_{0}). So the curve l|_{[ε,1]} has endpoints in Λ\Λ_{0} and determines an element in H_{1}(Λ, Λ\Λ_{0}; ℤ), which for the last item above, is isomorphic to ℤ. The image of the homology class defined by l|_{[ε,1]} under the isomorphism H_{1}(Λ, Λ \ Λ_{0}; ℤ) ≅ ℤ is called the Maslov index of the Morse–Sturm system (6).

Remark 3.1 At the level of fundamental group, the above construction can be carried on as follows: take any curve l′ in Λ\Λ_{0} connecting l(ε) and l(1). The curves l|_{[ε,1]} and l′ define a loop in Λ whose homotopy class does not depend on the choice of l′ because Λ \ Λ_{0} is contractible. This construction produces a (well-defined) element of π_{1}(Λ) and hence an integer.

It is important to notice that the manifold Λ is orientable iff n is odd. Our approach to the Maslov index avoids the problem of using Poincaré duality in the non orientable case, hence cohomology with twisted coefficients (see [1, 6] among others). This approach works well since, in any case, the normal bundle to Λ_{1} (which is the regular part of Λ_{0}), is orientable, so we can develop a reasonable intersection theory.

If l : [0, 1] → Λ is a curve of class C^{1} and if l(t) ∈ Λ_{0} then we call l(t) a nondegenerate intersection of l with Λ_{0} if the symmetric bilinear form l′(t) : l(t) × l(t) → IR is nondegenerate on l(t) ∩ l_{0}.

In the following proposition it is given a method for computing the relative homology class of a differentiable curve in Λ in a generic situation:

Proposition 3.2 If l : [0, 1] → Λ is a C^{1} curve with endpoints in Λ \ Λ_{0} having only nondegenerate intersections with Λ_{0} then the integer corresponding to the homology class of l in H_{1}(Λ, Λ\Λ_{0}) is equal to the sum ∑_{t∈(0,1)} sgn(l′(t)|_{l(t)∩l_{0}}).

We have now our way paved to give a sketchy proof of Theorem 2.6, as follows. Consider the eigenvalue problem:

−J′′(t) + R_{λ}(t)J(t) = 0,  (8)

where R_{λ}(t) = R(t) − λ; clearly, for λ = 0 the system (8) becomes (6). For each λ ∈ IR we have an associated curve of Lagrangians l_{λ} : [0, 1] → Λ; this defines a differentiable map (t, λ) ↦ l(t, λ) ∈ Λ. The following facts are proven in [10]:

1. for λ < −∥R∥_{∞} the system (8) has no conjugate instants;

2. there exists ε > 0 such that the system (8) has no conjugate instants in (0, ε] for all λ ∈ [−∥R∥_{∞}, 0];

3. if t ∈ (0, 1] is a conjugate instant of (8) with λ = 0 then l(t)∩l_{0} = {0}⊕𝕁[t]^{⊥} and l′(t)|_{l(t)∩l_{0}} is identified with the restriction of ⟨·, ·⟩ to 𝕁[t]^{⊥}; 4. λ ∈ IR is an eigenvalue of J iff ℓ(1, λ) ∈ Λ_{0}; moreover, the map J ↦ (J(1), J′(1)) defines an isomorphism from the eigenspace H_{λ} and the intersection l_{λ}(1) ∩ l_{0} that carries the restriction of g̃ to the restriction of \frac{∂}{∂λ}l_{λ}(1).

Keeping in mind Proposition 3.2, from facts 1, 3 and 4 we get immediately the following:

Proposition 3.3 If all the conjugate instants are nondegenerate and t = 1 is non conjugate, then the Maslov index of the system (6) coincides with the geometric index i_{geom}. Moreover, if all the real negative eigenvalues of J are nondegenerate, then the homology class of the curve [−∥R∥_{∞}, 0] ∋ λ ↦ l(1, λ) is equal to the spectral index i_{Morse}.

Consider the image by the map (t, λ) ↦ ℓ(t, λ) of the rectangle

[ε, 1] × [−∥R∥_{∞}, 0]

shown in the figure below.

Assume that t = 1 is non conjugate, that all the conjugate instants of (6) are nondegenerate and that all real negative eigenvalues of J are nondegenerate. By Proposition 3.3, the relative homology class of the image by l of α_{2} equals the geometric index i_{geom} and the relative homology class of the image by l of α_{3} equals the spectral index i_{Morse}. Fact 1 implies that α_{4} is mapped by l into the complement of Λ_{0} and Fact 2 implies that α_{2} is mapped by l into the complement of Λ_{0}. Since the chain α_{3} − α_{2} − α_{1} + α_{4} is homologous to zero in the rectangle [−∥R∥_{∞}, 0] × [ε, 1], it follows that the images of α_{3} and α_{2} by l define the same homology class in H_{1}(Λ, Λ \ Λ_{0}). This concludes the proof of Theorem 2.6.

[figure: ▶; t; ▲; λ; >; α_{2}; 0; ε; ∧; α_{1}; 1; ∧; α_{3}; −∥R∥_{∞}; >; α_{4}]

4 On the stability of the geometric index

From Proposition 3.3 and the topological character of the Maslov index, we obtain the following generalization of Corollary 1.3:

Theorem 4.1 Let γ_{n} : [0, 1] → M be a sequence of geodesics converging to a geodesic γ : [0, 1] → M on a semi-Riemannian manifold M. Assume that all the conjugate instants along γ and along all the γ_{n}’s are nondegenerate. Then if t = 1 is non conjugate along γ we have

lim_{n→∞} i_{geom}(γ_{n}) = i_{geom}(γ).

Remark 4.2 Observe that Theorem 4.1 does not imply that the set of conjugate instants is stable. For instance, it is quite possible that a conjugate instant of signature zero may “evaporate” by a slight perturbation of the geodesic.

Remark 4.3 If a geodesic has degenerate conjugate instants then it is indeed possible that its Maslov index differs from the geometric index. An explicit example of this kind (in the real-analytic case) is given in the last section of [10]. Observe that while the Maslov index is always stable, the geometric index may not be stable in the presence of degenerate conjugate instants.

Remark 4.4 If γ is a causal (i.e., non spacelike) geodesic in a Lorentzian manifold, it is easy to see that all conjugate instants are nondegenerate and that their signature equals their multiplicity. Hence, Theorem 4.1 implies that if a sequence of causal geodesics γ_{n} converges to a (necessarily causal) geodesic γ in a Lorentzian manifold then lim_{n→∞} i_{geom}(γ_{n}) = i_{geom}(γ), provided that t = 1 is non conjugate along γ. This fact was used in [7] to develop a Morse theory for lightlike geodesic in Lorentzian manifolds as a “limit” of a Morse theory for timelike geodesics.

References

[1] V. I. Arnol’d, Characteristic Class Entering in Quantization Conditions, Funct. Anal. Appl. 1 (1967), 1–13.

[2] Glen E. Bredon, Topology and Geometry, Springer–Verlag, Graduate Text in Mathematics 139.

[3] M. do Carmo, Riemannian Geometry, Birkhäuser, Boston, 1992.

[4] J. C. Corrêa Eidam, A. L. Pereira, P. Piccione, D. V. Tausk, On the Equality between the Maslov Index and the Spectral Index for the semi-Riemannian Jacobi Operator, preprint 2000.

[5] J. J. Duistermaat, On the Morse Index in Variational Calculus, Adv. in Math. 21 (1976), 173–195.

[6] D. B. Fuks, Maslov–Arnol’d Characteristic Classes, Soviet Math. Dokl. 9 (1968), No. 1, 96–99.

[7] F. Giannoni, A. Masiello, P. Piccione, A Morse Theory for Massive Particles and Photons by Fermat Principles in General Relativity Relativity, Journal of Geometry and Physics 35, vol. 1 (2000), 1–35.

[8] F. Giannoni, A. Masiello, P. Piccione, D. Tausk, A Generalized Index Theorem for Morse– Sturm Systems and Applications to semi-Riemannian Geometry, to appear in the Asian Journal of Mathematics (LANL math.DG/9908056).

[9] A. D. Helfer, Conjugate Points on Spacelike Geodesics or Pseudo-Self-Adjoint Morse-Sturm-Liouville Systems, Pacific J. Math. 164, n. 2 (1994), 321–340.

[10] F. Mercuri, P. Piccione, D. Tausk, Stability of the Focal and the Geometric Index in semi-Riemannian Geometry via the Maslov Index, Technical Report RT-MAT 99-08, Mathematics Department, University of São Paulo, Brazil, 1999. (LANL math.DG/9905096)

[11] P. Piccione, D. V. Tausk, An Index Theorem for Non Periodic Solutions of Hamiltonian Systems, Proceedings of the London Mathematical Society Society (3) 83 (2001), 351–389.

[12] P. Piccione, D. V. Tausk, The Maslov Index and a Generalized Morse Index Theorem for Non Positive Definite Metrics, Comptes Rendus de l’Académie de Sciences de Paris, vol. 331, 5 (2000), 385–389.

[13] P. Piccione, D. V. Tausk, On the Distribution of Conjugate Points along semi-Riemannian Geodesics, preprint 2000, to appear in Communications in Analysis and Geometry (LANL

math.DG/0011038).