Off-campus UMass Amherst users: To download dissertations, please use the following link to log into our proxy server with your UMass Amherst user name and password.
Non-UMass Amherst users, please click the view more button below to purchase a copy of this dissertation from Proquest.
(Some titles may also be available free of charge in our Open Access Dissertation Collection, so please check there first.)
Compact Lie group actions and total mean curvatures of irreducible symmetric subspaces
Abstract
This dissertation consists of three parts: "Compact Lie group actions on aspherical $A\sb{k}(\pi)$-manifolds", "Total mean curvature of symmetric subspaces of symmetric subspaces of Euclidean spaces" and "Discrete Wirtinger and isoperimetric-type inequalities". In the first part, let M be an aspherical $A\sb{k}(\pi)$-manifold and $\pi\prime$ torsion-free, where $\pi\prime$ is some quotient group of $\pi$. We prove that: (1) Suppose the Euler characteristic $\chi(M) \not=$ 0. If G is compact Lie group acting effectively on M, then G is a finite group, (2) $N\sbsp{T}{s}\leq (n-k)(n-k$ + 1)/2, where $N\sbsp{T}{s}(M)$ denotes the semisimple degree of symmetry of M. We are also able to unify many well-known results with simpler proofs. In the second part, let $M\sp{n}$ = G/K be an irreducible symmetric subspace of $R\sp{m}$ and x: $M \to R\sp{m}$ be an equivariant isometric immersion. We obtain some good lower bound and upper bound estimates of various total mean curvatures of these submanifolds which improve or generalize some well-known results. We improve a result of T. Takahashi by proving the following: let $M\sp{n}$ = G/K and x be as above. If the isotropy action is transitive on the unit sphere of the tangent space $T\sb{eK}M$ of M at base point eK. Then $x = x\sb0 + a\sb1x\sb1 {+}\cdots{+}\ a\sb{r}x\sb{r},$ with $\sum\sbsp{i = 1}{r}\ a\sbsp{i}{2} = 1,$ and the eigenfunctions $x\sb{i} = (\phi\sb1,\cdots,\phi\sb{n\sb i})$ form a basis of eigenspace for $\lambda\sb{i}$. We have a good lower bound estimate for the integral of the k-th power of the length of the second fundamental form which improves a result of D. Bleeker and J. Weiner (6) if $M\sp{n}$ is as above. Finally we prove some new, interesting asymptotic formulas concerning various total mean curvatures, for instance, we have$$\lim\sb{k\to\infty}\ {\int\sb{M}\vert H\sb{k}\vert\sp{n}dV\sb{g}\over k} = {(2\pi)\sp{n}\over n\sp{n/2}\cdot w\sb{n}},$$where $w\sb{n}$ is the volume of the unit disk of $R\sp{n}$ and $H\sb{k}$ is the mean curvature of k-th standard immersion. The last part of the dissertation is devoted to the investigation of some new Wirtinger-type and new isoperimetric-type inequalities. Our main result is the following: if $f(\theta)$ is a $C\sp2$-function on (0,l) and $f(\theta)$ also satisfies some natural conditions, then we have: $$\lbrack{\sum\limits\sbsp{j = 1}{n}} f(\theta\sb{j})\rbrack\sp2 \geq C\sb{n} {\sum\limits\sbsp{j = 1}{n}} f(\theta\sb{j}) \cdot f\sp\prime(\theta\sb{j}),\quad 0 < \theta\sb{j} < l\quad j = 1,\cdots,n,$$where $\sum\sbsp{j = 1}{n} \theta\sb{j}$ = $ml, 0 < m < n,$ and $C\sb{n}$ is a constant. Equality holds if and only if $\theta\sb1\ {=}\cdots{=}\ \theta\sb{n}$. We obtain some new isoperimetric-type inequalities from above inequality.
Subject Area
Mathematics
Recommended Citation
Tang, Dingyi, "Compact Lie group actions and total mean curvatures of irreducible symmetric subspaces" (1990). Doctoral Dissertations Available from Proquest. AAI9110221.
https://scholarworks.umass.edu/dissertations/AAI9110221