Related papers: The Geometry of Loop Spaces II: Corrections

We consider the Lie group of smooth diffeomorphisms Diff$(M)$ of a simple polytope $M$ in the euclidean space. Simple polytopes are special cases of manifolds with corners. The geometric setting allows to study in particular, the subgroup…

This paper is the next installment of our analysis of length-commensurable locally symmetric spaces begun in Publ. math. IHES 109(2009), 113-184. For a Riemannian manifold $M$, we let $L(M)$ be the weak length spectrum of $M$, i.e. the set…

We study homogeneous metric spaces, by which we mean connected, locally compact metric spaces whose isometry group acts transitively. After a review of some classical results, we use the Gleason-Iwasawa-Montgomery-Yamabe-Zippin structure…

The first section of this modest survey reviews some basic notions and describes some families of examples, and the second section briefly indicates some general aspects of analysis on metric spaces. The remaining three sections are…

Two smooth manifolds M and N are called R-diffeomorphic if their product with the real line are diffeomorphic. We consider the following simplification problem: does R-diffeomorphism imply diffeomorphism or homeomorphism? For compact…

We discuss Lie-Tresse theorem for the pseudogroup of diffeomorphisms acting on the space of (pseudo-)Riemannian metrics, and relate this to existence of Killing vector fields. Then we discuss the impact of symmetry in the general case.

A consistent approach to the description of integral coordinate invariant functionals of the metric on manifolds ${\cal M}_{\alpha}$ with conical defects (or singularities) of the topology $C_{\alpha}\times\Sigma$ is developed. According to…

We develop sheaf theory in the context of difference algebraic geometry. We introduce categories of difference sheaves and develop the appropriate cohomology theories. As specializations, we get difference Galois cohomology, difference…

We work on a 4-manifold equipped with Lorentzian metric $g$ and consider a volume-preserving diffeomorphism which is the unknown quantity of our mathematical model. The diffeomorphism defines a second Lorentzian metric $h$, the pullback of…

This thesis consists of two parts, connected by one central theme: the dynamics of the "shape of space". The first part of the thesis concerns the construction of a theory of gravity dynamically equivalent to general relativity (GR) in 3+1…

Let $M$ be a closed orientable 3-manifold with scalar curvature greater than or equal to 1. If $M$ has nonvanishing second homotopy group, then it is known that the $\pi_2$-systole of $M$ (i.e. the minimal achievable area of homotopically…

Let M be a smooth compact connected oriented manifold of dimension at least two endowed with a volume form. Assuming certain conditions on the fundamental group $\pi_1(M)$ we construct quasi-isometric embeddings of either free Abelian or…

The main topic is the development of a Fredholm theory in a new class of spaces called M-polyfolds. In the subsequent Volume II the theory will be generalized to an even larger class of spaces called polyfolds, which can also incorporate…

We study two aspects of the loop group formulation for isometric immersions with flat normal bundle of space forms. The first aspect is to examine the loop group maps along different ranges of the loop parameter. This leads to various…

The present paper gives an account and quantifies the change in topology induced by small and type II geometric transitions, by introducing the notion of the \emph{homological type} of a geometric transition. The obtained results agree…

The paper is devoted to a categorical study of the category of probabilistic metric spaces. The study is based on an isomorphic description of the category of probabilistic metric spaces. The isomorphic description was obtained in [3] and…

We define and investigate $\alpha$-modulation spaces $M_{p,q}^{s,\alpha}(G)$ associated to a step two stratified Lie group $G$ with rational structure constants. This is an extension of the Euclidean $\alpha$-modulation spaces…

We carry out analysis and geometry on a marked configuration space $\Omega_X^{R_+}$ over a Riemannian manifold $X$ with marks from the space $R_+$ as a natural generalization of the work {\bf [}{\it J. Func. Anal}. {\bf 154} (1998),…

This paper is concerned with analysis on metric spaces in a variety of settings and with several kinds of structure.

Motivated by the computation of loop space quantum mechanics as indicated in [7], here we seek a better understanding of the tubular geometry of loop space ${\cal L}{\cal M}$ corresponding to a Riemannian manifold ${\cal M}$ around the…