where g(t) is a family of Kähler metrics and Ric(ωg) denotes the Ricci curvature of g. It is known that for any smooth Kähler metric g0, there is a unique solution g(t) of (1.1) for some maximal time T > 0 with g(0) = g0. In general, T will depend on the initial metric g0. However, in Kähler manifold, this only depends on the Kähler class and the first Chern class. This observation plays a impo...

In this note we discuss conditions under which a linear connection on a manifold equipped with both a symmetric (Riemannian) and a skew-symmetric (almost-symplectic or Poisson) tensor field will preserve both structures. If (M, g) is a (pseudo-)Riemannian manifold, then classical results due to T. Levi-Civita, H. Weyl and E. Cartan [7] show that for any (1, 2) tensor field T i jk which is skew-...

Inspired by Gromov's work on 'Metric inequalities with scalar curvature' we establish band width for Riemannian bands of the form $(V=M\times[0,1],g)$, where $M^{n-1}$ is a closed manifold. We introduce new class orientable manifolds call filling enlargeable and prove: If $M$ all unit balls in universal cover $(V,g)$ have volume less than constant $\frac{1}{2}\varepsilon_n$, then $width(V,g)\le...

In this paper we will determine the multiple point manifolds of certain self-transverse immersions in Euclidean spaces. Following the triple points, these immersions have a double point self-intersection set which is the image of an immersion of a smooth 5-dimensional manifold, cobordant to Dold manifold $V^5$ or a boundary. We will show there is an immersion of $S^7times P^2$ in $mathbb{R}^{1...

Lickorish has constructed large families of contractible 4–manifolds that have knotted embeddings in the 4–sphere and has also shown that every finitely presented perfect group with balanced presentation occurs as the fundamental group of the complement of a knotted contractible manifold. Here we make a few observations regarding Lickorish’s construction, showing how to extend it to construct c...

In this work, we obtain a short-time existence result for harmonic map heat flow coupled with smooth family of complete metrics in the domain manifold. Our results generalize by Li-Tam (Invent Math 105(1):1–46, 1991) and Chen-Zhu (J Differ Geom 74:119–154, 2006). particular, prove along Ricci g(t) on M into manifold curvature bounded from above, under assumption: (i) $$|\text {Rm}(g(t))|\le a/t...

Bordism theories are particular sorts of homology theories, built in accordance to the rule singular homology : simplices :: bordism : manifolds. Fixing a structure group G, the geometric chain complex GC∗(X ;G) of a space X is given by GCn(X ;G) =N f : M →X M a connected manifold with G-structure . The boundary maps in this complex are induced by the restriction of f to the boundary of M . Hen...

commutative properties of four-dimensional generalized symmetric pseudo-riemannian manifolds were considered. specially, in this paper, we studied skew-tsankov and jacobi-tsankov conditions in 4-dimensional pseudo-riemannian generalized symmetric manifolds.

in this paper, we obtain a necessary and sufficient condition for a conformal mapping between two weyl manifolds to preserve einstein tensor. then we prove that some basic curvature tensors of $w_n$ are preserved by such a conformal mapping if and only if the covector field of the mapping is locally a gradient. also, we obtained the relation between the scalar curvatures of the weyl manifolds r...

Given a complex manifold M equipped with a holomorphic action of a connected complex Lie group G, and a holomorphic principal H–bundle EH over X equipped with a G–connection h, we investigate the connections on the principal H–bundle EH that are (strongly) adapted to h. Examples are provided by holomorphic principal H– bundles equipped with a flat partial connection over a foliated manifold.