Tubular presentations pi of subsets of manifolds

In this note the Weierstrass integral J is taken as L, the Riemann integral of length. Here M(n), n > 1, is a compact, connected manifold of class C(infinity) with a positive definite Riemann structure. Presentations (phi, U(phi)) in M(n) and geodesics (with the aid of the Euler-Riemann equations) are defined in Morse, M. (1976) "Global variational analysis: Weierstrass integrals on a Riemannian Manifold," Mathematical Notes (Princeton University Press, Princeton, NJ). Christoffel terminology is not suitable for our purposes. Conjugate points on a geodesic need a definition that does not distinguish between simple and nonsimple geodesics. Tubular presentations pi of an open neighborhood of a geodesic lead to such a definition. These presentations are locally biunique and "compatible" with the earlier biunique presentations in M(n). We shall use a presentation pi in proving Theorem A below. Theorem A is essential in computing the Fréchet numbers of product manifolds as we shall show in a later paper.