Let M̃ be a (2m+ 1)-dimensional almost contact manifold with almost contact structure (φ,ξ,η), that is, a global vector field ξ, a (1,1) tensor field φ, and a 1-form η on M̃ such that φ2X =−X +η(X)ξ, η(ξ) = 1 for any vector field X on M̃. We consider a product manifold M̃×R, whereR denotes a real line. Then a vector field on M̃×R is given by (X , f (d/dt)), where X is a vector field tangent to M̃, t ...

Let $M$ be a smooth manifold. When $\Gamma$ is group acting on the manifold by diffeomorphisms one can define $\Gamma$-co-invariant cohomology of to differential complex $\Omega_c(M)_\Gamma=\mathrm{span}\{\omega-\gamma^*\omega,\;\omega\in\Omega_c(M),\;\gamma\in\Gamma\}.$ For Lie algebra $\mathcal{G}$ $M$, defines $\mathcal{G}$-divergence forms $\mathcal{C}_{\mathcal{G}}(M)=\mathrm{span}\{L_X\om...

We prove two lower bounds for the volumes of balls in a Riemannian manifold. If (Mn, g) is a complete Riemannian manifold with filling radius at least R, then it contains a ball of radius R and volume at least δ(n)Rn . If (Mn, hyp) is a closed hyperbolic manifold and if g is another metric on M with volume at most δ(n)V ol(M,hyp), then the universal cover of (M, g) contains a unit ball with vol...

Let ( X , J ) be an almost complex manifold with a (smooth) involution ? : ? such that Fix ? ? . Assume is conjugation, i.e., the differential of anti-commutes The space P m = S × / ? where v x ? known as generalized Dold manifold. Suppose group G ? Z 2 s acts smoothly on g ? for all ? Using action diagonal subgroup D O 1 + ? sphere we obtain which descends to When stationary point set finite, ...

The purpose of this note is to prove the following: Theorem 1.1 Let G be a nite group. Then there is a rational homology S 3 on which G acts freely. That any nite group acts freely on some closed 3-manifold is easy to arrange: There are many examples of closed 3-manifolds whose fundamental groups surject a free group of rank two (for example, by taking a connected sum of S 1 S 2 's) and by pass...

zw Let G be a finite abelian group and M an o-minimal expansion of Rexp = (R, +, ·, <, e) admitting the C∞ cell decomposition. Everything is considered in M. We prove that every definable C∞G manifold is affine. Moreover we prove that if X1, . . . , Xn (resp. Y1, . . . , Yn) are definable C ∞G submanifolds of a definable C∞G manifold X (resp. Y ) in general position, then every definable CG map...

In this note under a crucial technical assumption we derive a differential equality of Yamabe constant Y (g (t)) where g (t) is a solution of the Ricci flow on a closed n-manifold. As an application we show that when g (0) is a Yamabe metric at time t = 0 and Rgα n−1 is not a positive eigenvalue of the Laplacian ∆gα for any Yamabe metric gα in the conformal class [g0], then d dt ∣∣ t=0 Y (g (t)...

Let g(t) be a family of smooth Riemannian metrics on an n-dimensional closed manifold M . Moreover, given a smooth closed Riemannian manifold (N, gN ) of arbitrary dimension, let φ(t) be a family of smooth maps from M to N . Then (g(t), φ(t)) is called a solution of the volume preserving Harmonic Ricci Flow (or Ricci Flow coupled with Harmonic Map Heat Flow), if it satisfies  ∂tg = −2 Ricg + ...

In the paper we give a partial answer to the following question: let G be a finite group acting smoothly on a compact (smooth) manifold M , such that for each isotropy subgroup H of G the submanifold M fixed by H can be deformed without fixed points; is it true that then M can be deformed without fixed points G-equivariantly? The answer is no, in general. It is yes, for any G-manifold, if and o...