Conjugate points on a limiting extremal

Abstract

Theorem 1, together with the author's extensions of the Sturmian Comparison Theorems, will suffice to establish the basic Theorem 27.3, p. 211 of Global Variational Analysis: Weierstrass Integrals on a Riemannian Manifold. Mathematical Notes, Princeton University Press, Princeton, N.J., 1976. Here and in the preceding reference there is given a Weierstrass Integral on a compact, connected, Riemannian manifold M(n), conditioned as in the above reference. Let gamma be an arbitrary extremal of J joining a point P to a point Q, not equal P on M(n). An extremal zeta of J joining A to B is termed nondegenerate if A and B are not conjugate on zeta. The index of zeta is by definition the "count" of conjugate points of A on zeta definitely preceding B.THEOREM 1. When the extremal gamma is nondegenerate, extremals of J issuing from P with the same length as gamma and sufficiently near gamma in the sense of Fréchet, are nondegenerate and have the same index as gamma.