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Menninga, Nadine Louise 2014-12-16T06:18:10Z 2014-12-16T06:18:10Z 10000-01-01 1984 1984 This paper investigates the following questions. If a compact manifold, M, with positive sectional curvature is isometrically immersed in some ambient space, N, what is the radius of the smallest ball in which its image lies? Additionally, when can the knowledge that the image lies inside a ball of restricted size be used to conclude that the immersion is an imbedding whose image bounds a convex body in N? The major results presented in this paper are the following theorems: Theorem 1. Let M be a compact, connected Riemannian manifold of dimension m, m (GREATERTHEQ) 2, with K(M) (GREATERTHEQ) 1/c('2), where c is a positive constant. Let N be an n-dimensional Riemannian manifold such that (pi)c (LESSTHEQ) i(N) and K(N) ) N is an isometric immersion, then x(M) is contained in a metric ball of N with radius R 0 and simply connected space N of constant curvature less than 1/(4c('2)), there exists an M satisfying the above conditions and an immersion x: M (--->) M such that x(M) lies in no ball of radius 1/2(pi)c - (epsilon). Theorem 2. Let M be a compact, connected, orientable, Riemannian manifold of dimension n-1, with n-1 (GREATERTHEQ) 2, and K(M) (GREATERTHEQ) 1/c('2), c is a positive constant. Let N be an n-dimensional Riemannian manifold such that (pi)c (LESSTHEQ) i(N) and K(N) ) N be an isometric immersion. Then x imbeds M into N as the boundary of a convex body. In the above theorems K(N) is the sectional curvature function of the manifold N and i(N) is the injectivity radius of the manifold N. Made available in DSpace on 2014-12-16T06:18:10Z (GMT). No. of bitstreams: 1 8422134.pdf: 1124133 bytes, checksum: 4abb6cea43187b9cf419100ff763e09d (MD5) Previous issue date: 1984 Embargo set by: Seth Robbins for item 71386 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs U of I Only 40 p. Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1984. http://hdl.handle.net/2142/71220 (UMI)AAI8422134 Mathematics Immersions of Positively Curved Manifolds Into Manifolds With Curvature Bounded Above text Mathematics Mathematics University of Illinois at Urbana-Champaign Dissertation Ph.D.