Differential Geometry

It is proved that the Lie groups $\E_7^{(5)}$ and $\E^{(7)}_7$ represented in $\R^{56}$ and the Lie group $\E_7^{\C}$ represented in $\R^{112}$ occur as holonomies of torsion-free affine connections. It is also shown that the moduli spaces of torsion-free affine connnections with these holonomies are finite dimensional, and that every such connection has a local symmetry group of positive dimension.

Let Y be a CW-complex with a single 0-cell, let K be its Kan group, a free simplicial group whose realization is a model for the space ΩY of based loops on Y , and let G be a Lie group, not necessarily connected. By means of simplicial techniques involving fundamental results of {\smc Kan's} and the standard W - and bar constructions, we obtain a weak G -equivariant homotopy equivalence from the geometric realization $|\roman{Hom}(K,G)|$ of the cosimplicial manifold $\roman{Hom}(K,G)$ of homomorphisms from K to G to the space $\roman{Map}^o(Y,BG)$ of based maps from Y to the classifying space BG of G where G acts on BG by conjugation. Thus when Y is a smooth manifold, the universal bundle on BG being endowed with a universal connection, the space $|\roman{Hom}(K,G)|$ may be viewed as a model for the space of based gauge equivalence classes of connections on Y for all topological types of G -bundles on Y thereby yielding a rigorous approach to lattice gauge theory; this is illustrated in low dimensions.

Let Y be a CW-complex with a single 0-cell, K its Kan group, a model for the loop space of Y , and let G be a compact, connected Lie group. We give an explicit finite dimensional construction of generators of the equivariant cohomology of the geometric realization of the cosimplicial manifold $\roman{Hom}(K,G)$ and hence of the space $\roman{Map}^o(Y,BG)$ of based maps from Y to the classifying space BG . For a smooth manifold Y , this may be viewed as a rigorous approach to lattice gauge theory, and we show that it then yields, (i) when {$\roman{dim}(Y)=2$,} equivariant de Rham representatives of generators of the equivariant cohomology of twisted representation spaces of the fundamental group of a closed surface including generators for moduli spaces of semi stable holomorphic vector bundles on complex curves so that, in particular, the known structure of a stratified symplectic space results; (ii) when {$\roman{dim}(Y)=3$,} equivariant cohomology generators including the Chern-Simons function; (iii) when {$\roman{dim}(Y) = 4$,} the generators of the relevant equivariant cohomology from which for example Donaldson polynomials are obtained by evaluation against suitable fundamental classes corresponding to moduli spaces of ASD connections.

A Chern-Weil construction for extensions of Lie-Rinehart algebras is introduced. This generalizes the classical Chern-Weil construction in differential geometry and yields characteristic classes for arbitrary extensions of Lie-Rinehart algebras. Some examples arising from spaces with singularities and from foliations are given that cannot be treated by means of the classical Chern-Weil construction.

This is an elementary introduction to exterior differential systems motivated by two examples: minimal submanifolds and the isometric embedding problem. The two main goals of the lectures are: 1. To explain how to find an appropriate geometric setting for studying a given system of pde. 2. To explain the Cartan algorithm to determine the moduli space of local solutions to any given exterior differential system and an appropriate initial value problem for the system.

Let Z be a compact complex (2n+1)-manifold which carries a {\em complex contact structure}, meaning a codimension-1 holomorphic sub-bundle D of TZ which is maximally non-integrable. If Z admits a Kähler-Einstein metric of positive scalar curvature, we show that it is the Salamon twistor space of a quaternion-Kähler manifold (M^{4n}, g). If Z also admits a second complex contact structure, then Z= CP_{2n+1}. As an application, we give several new characterizations of the Riemannian manifold HP_n=Sp(n+1)/( Sp(n)\times Sp(1)).

In this paper we study geometry of symmetric torsion-free connections which preserve a given symplectic form

In a noncompact harmonic manifold M we establish finite dimensionality of the eigenspaces V λ generated by radial eigenfunctions of the form coshr+c . As a consequence, for such harmonic manifolds, we give an isometric imbedding of M into ( V λ ,B) , where B is a nondegenerate symmetric bilinear indefinite form on V λ (analogous to the imbedding of the real hyperbolic space I H n into I R n+1 with the indefinite form Q(x,x)=− x 2 0 +∑ x 2 i ). Finally we give certain conditions under which M is symmetric.

We corrected a few errors in the previous submission. These do not affect any of the topological conclusions of the earlier version. We have also included a few observations about the Casson invariant of the Brieskorn homology spheres.

An anologue of the Calabi invariant for Poisson manifolds is considered. For any Poisson manifold P , the Poisson bracket on C ∞ (P) extends to a Lie bracket on the space Ω 1 (P) of all differential one-forms, under which the space Z 1 (P) of closed one-forms and the space B 1 (P) of exact one-forms are Lie subalgebras. These Lie algebras are related by the exact sequence: 0\lon \reals \lon C^{\infty}(P)\stackrel{d}{\lon} Z^{1}(P)\stackrel{f}{\lon} H^{1}(P, \reals)\lon 0, where $H^{1}(P,\reals)$ is considered as a trivial Lie algebra, and f is the map sending each closed one-form to its cohomology class. The goal of the present paper is to lift this exact sequence to the group level for compact Poisson manifolds under certain integrability condition. In particular, we will give a geometric description of a Lie group integrating the underlying Poisson algebra C ∞ (P) .