The Conjugate Locus of a Riemannian Manifold

Abstract

Introduction. The conjugate locus (considered as a subset of the tangeit space to a point of a Rliemannian mauiifold) splits naturally into two subsetsthe regular locus and the singular locus; the latter, roughly speaking, consists of those points which occur at intersections in the conjugate locus. We describe in this paper the regular conjugate locus and the nature of the exponential map nearby. Our main results are these: The regular conjugate locus is dense in the conjugate locus and is a submanifold of the tangent space of codimension one. If the order of a regular conjugate point, i. e., the dimension of the kernel of the differential of the exponential map there, is greater than or equal to 2, the kernel must actually be tangent to the regular conjugate locus. This was proved for analytic Finsler spaces by J. HI. C. Whitehead in [11] in the case that the order of the conjugate point is greater than half the dimension of the manifold. We obtaini normal forms for the exponential map on neighborhoods of all regular conjugate points except for certain of the order 1 cases. As a corollary we obtain a new proof