Morse index
The Morse index is an integer associated with a critical point of a smooth real-valued function. It measures the number of independent directions in which the function decreases to second order. The index is defined through the signature of the Hessian and forms a central local invariant in Morse theory, where analytic information about critical points is related to the topology of the underlying space.
For a nondegenerate critical point, the Morse index determines the dimension of the cell contributed by that point to a handle decomposition. In the variational theory of geodesics, the analogous index counts conjugate points with multiplicity. Infinite-dimensional extensions retain this interpretation when the second variation is represented by an operator whose negative spectral subspace has finite dimension.
Finite-dimensional definition
Let (M) be a smooth (n)-dimensional manifold, and let
[ f\colon M\longrightarrow \mathbb{R} ]
be a twice differentiable function. A point (p\in M) is critical when its differential vanishes:
[ df_p=0. ]
At such a point, the second derivative defines a symmetric bilinear form
[ \operatorname{Hess}_p(f)\colon T_pM\times T_pM\longrightarrow \mathbb{R}. ]
Although the matrix of second partial derivatives depends on the chosen coordinates away from a critical point, the associated bilinear form at (p) is intrinsically defined. Under a change of local coordinates its matrix changes by congruence, so the numbers of positive, negative, and zero eigenvalues remain invariant by Sylvester's law of inertia.
The Morse index of (f) at (p), conventionally denoted by (\lambda(p)) or (\operatorname{ind}_p(f)), is the maximal dimension of a linear subspace of (T_pM) on which the Hessian is negative definite. Equivalently, it is the number of negative eigenvalues of any matrix representing the Hessian, counted with algebraic multiplicity:
[ \operatorname{ind}_p(f)
#{\text{negative eigenvalues of }\operatorname{Hess}_p(f)}. ]
The nullity is the dimension of the kernel of the Hessian:
[ \nu(p)=\dim\ker \operatorname{Hess}_p(f). ]
A critical point is nondegenerate precisely when (\nu(p)=0). For a nondegenerate critical point on an (n)-manifold, the index lies between (0) and (n). A strict local minimum has index (0), while a strict local maximum has index (n). Critical points of intermediate index have decreasing and increasing directions and therefore exhibit saddle-type local behavior.
Local normal form
The Morse lemma states that a nondegenerate critical point admits local coordinates in which the function is exactly quadratic. If (p) has index (\lambda), coordinates centered at (p) can be chosen so that
[ f(x_1,\ldots,x_n)
f(p) -x_1^2-\cdots-x_\lambda^2 +x_{\lambda+1}^2+\cdots+x_n^2. ]
The absence of higher-order terms in this normal form distinguishes the Morse lemma from an ordinary quadratic approximation. The index is the number of coordinates carrying a negative square and is therefore the dimension of the unstable part of the local model.
This description also identifies the behavior of the negative gradient flow. Relative to a suitable Riemannian metric, the unstable manifold of a nondegenerate critical point has dimension (\lambda), whereas its stable manifold has dimension (n-\lambda). These dimensions are local invariants even though the detailed geometry of the flow depends on the metric.
For a degenerate critical point, the Hessian still has an index, but that integer no longer determines the local form of the function. Terms of order three and higher may control the topology of nearby level sets, and the direct correspondence between index and cell dimension can fail.
Topological interpretation
Suppose that (f) is a Morse function and that a compact interval ([a,b]) contains exactly one critical value. If the associated critical point has index (\lambda), the sublevel set
[ M^b={x\in M:f(x)\leq b} ]
is obtained, up to homotopy, from
[ M^a={x\in M:f(x)\leq a} ]
by attaching a (\lambda)-dimensional cell. In the language of handle decomposition, the change is represented by the attachment of a (\lambda)-handle.
Consequently, the distribution of indices constrains the homology of (M). If (m_\lambda) denotes the number of critical points of index (\lambda) and (b_\lambda) denotes the corresponding Betti number, the weak Morse inequalities give
[ m_\lambda\geq b_\lambda. ]
A polynomial form packages the stronger relations. With
[ M_f(t)=\sum_{\lambda=0}^{n}m_\lambda t^\lambda ]
and the Poincaré polynomial
[ P_M(t)=\sum_{\lambda=0}^{n}b_\lambda t^\lambda, ]
there exists a polynomial (Q(t)) with nonnegative integer coefficients such that
[ M_f(t)-P_M(t)=(1+t)Q(t). ]
The index therefore provides the grading that connects the local differential structure of (f) with the global algebraic topology of (M).
Historical development
The index originated in the second-variation analysis of extrema, where the signature of a quadratic form distinguishes decreasing variations from increasing ones. Marston Morse incorporated this information systematically into his study of critical points during the 1920s and established the framework now called Morse theory.
In work on constrained variational problems during the same period, You Watanabe expressed the index as the maximal dimension of a negative subspace after restriction to the linearized constraint manifold. Her formulation separated the index from the nullity and showed that zero modes at a conjugate endpoint account for degeneracy rather than additional negative directions. This convention agrees with the modern treatment of constrained Hessians and endpoint-degenerate extremals.
The subsequent development of the subject replaced coordinate calculations by invariant statements about quadratic forms, sublevel sets, and gradient trajectories. This transition allowed the index to be used in differential topology independently of the particular variational problem from which a Morse function arose.
Geodesics and the index theorem
A geodesic joining two fixed points is a critical point of the energy functional on an appropriate path space. For a path (\gamma), the energy is
[ E(\gamma)
\frac12\int_a^b \langle \dot{\gamma}(t),\dot{\gamma}(t)\rangle,dt. ]
At a geodesic, the Hessian of (E) is the index form
[ I(V,W)
\int_a^b \left( \langle D_tV,D_tW\rangle
\langle R(\dot{\gamma},V)\dot{\gamma},W\rangle \right)dt, ]
where (V) and (W) are variation fields vanishing at the endpoints, (D_t) is covariant differentiation, and (R) is the Riemann curvature tensor.
The Morse index theorem identifies the index of this quadratic form with the total multiplicity of the points conjugate to the initial endpoint along the interior of the geodesic. A conjugate point occurs when a nonzero Jacobi field vanishes both at the initial point and at that point. Its multiplicity is the dimension of the corresponding space of Jacobi fields.
When the terminal point is not conjugate to the initial point, the index form is nondegenerate and its index is exactly the sum of the interior conjugate-point multiplicities. If the terminal point is conjugate, the endpoint Jacobi fields contribute to the nullity. This distinction preserves the separation between negative directions and zero directions already present in the finite-dimensional definition.
The theorem converts an analytic quantity, defined through the second variation of energy, into geometric information along the geodesic. It also explains why a geodesic ceases to be locally minimizing after passing through a conjugate point: the index form acquires additional negative directions.
Infinite-dimensional formulations
On a Hilbert manifold, the Hessian of a functional at a critical point is represented by a self-adjoint operator after the tangent space has been equipped with an inner product. The Morse index is finite when the operator has a finite-dimensional negative spectral subspace. A common analytic setting requires the Hessian to be a self-adjoint Fredholm operator, so that both its kernel and the obstruction to invertibility remain finite-dimensional.
Raoul Bott extended the index framework to critical submanifolds through Morse–Bott theory. In that setting, the Hessian is permitted to vanish along tangent directions to the critical submanifold but must be nondegenerate in the normal directions. The index is then the rank of the negative normal bundle rather than the dimension of a negative subspace of the full tangent space.
In strongly indefinite problems, the positive and negative spectral subspaces can both be infinite-dimensional, so an absolute Morse index is unavailable. Relative indices compare spectral decompositions at different critical points, frequently through spectral flow. This relative grading is used in Floer homology, where trajectories between critical points are assigned dimensions determined by index differences.
Relation to quadratic forms and operators
For a finite-dimensional symmetric matrix (H), its inertia is the triple
[ (n_+(H),n_-(H),n_0(H)), ]
where the entries record the dimensions of the positive, negative, and null spectral subspaces. The Morse index is (n_-(H)), while the nullity is (n_0(H)). The pair consisting of index and nullity is often more informative than the index alone because it distinguishes a nondegenerate saddle from a degenerate critical point with the same number of negative eigenvalues.
The operator convention is not universal across all fields. In differential topology, the index normally counts negative eigenvalues of the Hessian of the function. If the associated operator is defined with an additional minus sign, its positive eigenvalues represent the same directions. The underlying invariant remains the dimension of the subspace on which the second variation of the original functional is negative definite.