Conjugate points on a limiting extremal.
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.
Institute for Advanced Study, Princeton, New Jersey 08540.
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