We prove that, for an analytic family of “weakly tame” regular functions on an affine manifold, the spectrum at infinity of each function of the family is semicontinuous in the sense of Varchenko.

We show that the holonomy invariance of a function on the tangent bundle of a manifold, together with very mild regularity conditions on the function, is equivalent to the existence of local parallelisms compatible with the function in a natural way. Thus, in particular, we obtain a characterization of generalized Berwald manifolds. We also construct a simple example of a generalized Berwald ma...

We present a new proof of a strengthened version of D.V. Anosov's multiphase averaging theorem, originally stated for systems of ODEs with slow variables evolving in R and fast variables evolving on a smooth immersed manifold. Our result allows the fast variables to belong to an arbitrary smooth compact Riemannian manifold, and the vector eld to have only Sobolev regularity. This is accomplishe...

We revisit construction of the Atiyah-Hitchin manifold in generalized Legendre transform approach. This is originally studied by Ivanov and Rocek subsequently investigated more Ionas, latter which explicit forms K\"ahler potential metric are calculated. There a difference between former latter. In approach, constructed from contour integration one function with holomorphic coordinates. The choi...

We find the necessary and sufficient conditions for a sequential warped product manifold to be quasi-Einstein manifold. also investigate standard static space-time generalized Robertson-Walker of quasi-constant curvature.

We prove that a weakly convex almost Kähler 4-manifold contains a compact, non-constant J-holomorphic curve if the corresponding monopole invariant is not zero and if the corresponding line bundle is non-trivial.