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IT IS well-known that if a compact group acts differentiably on a differentiable manifold, then this group must preserve some Riemannian metric on this manifold. From this point of view, we shall discuss certain facts about group actions on manifolds. In 01, we study the group of isometries of a non-compact manifold. We discover that if the group is non-compact, the manifold has to split as a product of the Euclidean space and another manifold. This gives some information on problem 33 of [5]. It also gives some information on the group of biholomorphic transformations of a complex manifold whose Bergman metric is not trivial. Then we find a topological obstruction for a non-compact manifold to admit an infinite group to act freely and properly discontinuously. Namely, we prove that the natural map from the de Rham cohomology group with compact support to the de Rham cohomology group without compact support is trivial when the first group has finite dimension. In $2, we obtain some topological obstructions for group actions by looking at the complex of invariant differential forms. We prove, for example, that if a compact group acts on a compact manifold with non-zero Euler number, then wI A*** A ut+,= 0 for all closed invariant l-forms wl,..., Ok+ 1 with k 2 the codimension of the principle orbit.(It may be interesting to note that the vanishing is on the form level so that any secondary obstruction should also vanish.) We also prove a topological version of this theorem for circle actions. An interesting corollary is that if a manifold is the connected sum of a torus and a compact manifold with Euler number# 2, then it does not admit any circle actions

In this paper we study compact Sasaki manifolds in view of transverse K¨ahler geometry and extend some results in K¨ahler geometry to Sasaki manifolds. In particular we define integral invariants which obstruct the existence of transverse K¨ahler metric with harmonic Chern forms. The integral invariant f1 for the first Chern class case becomes an obstruction to the existence of transverse K¨ahler metric of constant scalar curvature. We prove the existence of transverse K¨ahler-Ricci solitons (or Sasaki-Ricci soliton) on compact toric Sasaki manifolds whose basic first Chern form of the normal bundle of the Reeb foliation is positive and the first Chern class of the contact bundle is trivial. We will further show that if S is a compact toric Sasaki manifold with the above assumption then by deforming the Reeb field we get a Sasaki-Einstein structure on S. As an application we obtain Sasaki-Einstein metrics on the U(1)-bundles associated with the canonical line bundles of toric Fano manifolds, including as a special case an irregular toric Sasaki-Einstein metrics on the unit circle bundle associated with the canonical bundle of the two-point blow-up of the complex projective plane.

In the study of geometric objects that arise naturally, the main tools are either groups or equations. In the rst case, powerful algebraic methods are available and enable one to solve many deep problems. While algebraic methods are still important in the second case, analytic methods play a dominant role, especially when the dening equations are transcendental. Indeed, even in the situation where the geometric object is homogeneous or algebraic, analytic methods often lead to important contributions. In this talk, we shall discuss a class of problems in differential geometry and the analytic methods that are involved in solving such problems.

We consider the quermassintegral preserving flow of closed \emph{h-convex} hypersurfaces in hyperbolic space with the speed given by any positive power of a smooth symmetric, strictly increasing, and homogeneous of degree one function $f$ of the principal curvatures which is inverse concave and has dual $f_*$ approaching zero on the boundary of the positive cone. We prove that if the initial hypersurface is \emph{h-convex}, then the solution of the flow becomes strictly \emph{h-convex} for $t>0$, the flow exists for all time and converges to a geodesic sphere exponentially in the smooth topology.