Compact Lie group actions and total mean curvatures of irreducible symmetric subspaces

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: $$\lbrack{\sum\limits\sbsp{j = 1}{n}} f(\theta\sb{j})\rbrack\sp2 \geq C\sb{n} {\sum\limits\sbsp{j = 1}{n}} f(\theta\sb{j}) \cdot f\sp\prime(\theta\sb{j}),\quad 0 \u3c \theta\sb{j} \u3c l\quad j = 1,\cdots,n,$$where $\sum\sbsp{j = 1}{n} \theta\sb{j}$ = $ml, 0 \u3c m \u3c n,$ 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