### A Neumann problem with the $q$-Laplacian on a solid torus in the critical of supercritical case.

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A lower estimate is proved for the number of critical orbits and critical values of a G-invariant C¹ function $f:{S}^{n}\to \mathbb{R}$, where G is a finite nontrivial group acting freely and orthogonally on ${\mathbb{R}}^{n+1}\phantom{\rule{4pt}{0ex}}0$. Neither Morse theory nor the minimax method is applied. The proofs are based on a general version of Borsuk’s Antipodal Theorem for equivariant maps of joins of G-sets.

For a locally symmetric space $M$, we define a compactification $M\cup M\left(\infty \right)$ which we call the “geodesic compactification”. It is constructed by adding limit points in $M\left(\infty \right)$ to certain geodesics in $M$. The geodesic compactification arises in other contexts. Two general constructions of Gromov for an ideal boundary of a Riemannian manifold give $M\left(\infty \right)$ for locally symmetric spaces. Moreover, $M\left(\infty \right)$ has a natural group theoretic construction using the Tits building. The geodesic compactification plays two fundamental roles in...

We study finite $G$-sets and their tensor product with Riemannian manifolds, and obtain results on isospectral quotients and covers. In particular, we show the following: If $M$ is a compact connected Riemannian manifold (or orbifold) whose fundamental group has a finite non-cyclic quotient, then $M$ has isospectral non-isometric covers.

In this paper we survey some recent results on rank one symmetric space.

This work presents a setting for the formulation of the mechanics of growing bodies. By the mechanics of growing bodies we mean a theory in which the material structure of the body does not remain fixed. Material points may be added or removed from the body.

This is a survey article based on the author’s Master thesis on affine representations of a gauge group. Most of the results presented here are well-known to differential geometers and physicists familiar with gauge theory. However, we hope this short systematic presentation offers a useful self-contained introduction to the subject.In the first part we present the construction of the group of motions of a Hilbert space and we explain the way in which it can be considered as a Lie group. The second...