Morse index

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A number associated with a critical point of a smooth function on a manifold or of a geodesic on a Riemannian (or Finsler) manifold.

1) The Morse index of a critical point of a smooth function on a manifold is equal, by definition, to the negative index of inertia of the Hessian of at (cf. Hessian of a function), that is, the dimension of the maximal subspace of the tangent space of at on which the Hessian is negative definite. This definition makes sense also for twice (Fréchet) differentiable functions on infinite-dimensional Banach spaces. The only difference is that the value is admissible for the index. In this case it is expedient to introduce the idea of the co-index of a critical point of as the positive index of inertia of the Hessian (the second Fréchet differential) of at .

2) Let and be smooth submanifolds of a complete Riemannian space . For a piecewise-smooth path with , , transversal to and at its end-points and , the analogue of a tangent space is the vector space of all piecewise-smooth vector fields along for which , . For any geodesic with , orthogonal at its end-points and to and , respectively, the second variation of the action functional (see Morse theory) defines a symmetric bilinear functional on (the analogue of the Hessian). The Morse index of the geodesic is equal, by definition, to the negative index of inertia of this functional. The null space of on (the set of at which for all ) consists exactly of the Jacobi fields (cf. Jacobi vector field) . If , the geodesic is called -degenerate, and is called the order of degeneracy of the geodesic.

The case when is a point is considered below. Let be the normal bundle to in and let be its fibre over . The restriction of the exponential mapping defines a mapping . A geodesic , , , is -degenerate if and only if the kernel of the differential of at is not null; in this connection, the dimension of the kernel is equal to the order of degeneracy of the geodesic . A point , , is called a focal point of along if the geodesic is -degenerate; the order of degeneracy of is called the multiplicity of the focal point . By the Sard theorem, the set of focal points has measure zero, so a typical geodesic is non-degenerate. If also consists of one point ( is not excluded), then a focal point is called adjoint to along . The Morse index theorem [1] asserts that the Morse index of a geodesic is finite and equal to the number of focal points of , , taking account of multiplicity.


[1] M. Morse, "The calculus of variations in the large" , Amer. Math. Soc. (1934)
[2] W. Ambrose, "The index theorem in Riemannian geometry" Ann. of Math. , 73 (1961) pp. 49–86


There is a natural generalization of the Morse index of geodesics to variational calculus, which runs as follows. Let be a real-valued smooth function on an open subset of and let be a smooth submanifold of . Let be the space of smooth curves for which the -jet lies in and . Then is a Banach manifold, on which one has the smooth functional

One then considers the Morse index of at critical curves ; it is finite if the Hessian of is positive definite at , , (Legendre's condition, cf. Legendre condition).


[a1] J.W. Milnor, "Morse theory" , Princeton Univ. Press (1963)
[a2] W. Klingenberg, "Riemannian geometry" , de Gruyter (1982) (Translated from German)
[a3] W. Klingenberg, "Lectures on closed geodesics" , Springer (1978)
How to Cite This Entry:
Morse index. M.M. PostnikovYu.B. Rudyak (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098