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Tubular presentations pi of subsets of manifolds.

Abstract：

: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.

journal_name

Proc Natl Acad Sci U S A

journal_title

Proceedings of the National Academy of Sciences of the United States of America

authors

Morse M

doi

10.1073/pnas.74.6.2209

subject

Has Abstract

pub_date

1977-06-01 00:00:00

pages

2209-10

issue

6

eissn

0027-8424

issn

1091-6490

journal_volume

74

pub_type

杂志文章

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