This paper is connected with the problem of describing path metric spaces which are homeomorphic to manifolds and biLipschitz homogeneous, i.e., whose biLipschitz homeomorphism group acts transitively. Our main result is the following. LetX = G/H be a homogeneous space of a Lie group G, and let d be a geodesic distance on X inducing the same topology. Suppose there exists a subgroup GS of G whi...

of the Dissertation Almost-Kähler Anti-Self-Dual Metrics by Inyoung Kim Doctor of Philosophy in Mathematics Stony Brook University 2014 We show the existence of strictly almost-Kähler anti-self-dual metrics on certain 4-manifolds by deforming a scalar-flat Kähler metric. On the other hand, we prove the non-existence of such metrics on certain other 4-manifolds by means of SeibergWitten theory. ...

(2) Complex structures provide almost complex structures by definition, and symplectic structures provide almost complex structures by linear algebra or by a Riemannian metric. A manifold which is almost complex but neither symplectic nor complex is CP#CP#CP. It is not complex because it does not fit into the Kodaira classification [14]. It is not symplectic because Taubes showed [20] that a sy...

In this paper, we investigate geodesics of the tangent bundle $TM$ a Riemannian manifold $(M,g)$ endowed with an arbitrary pseudo-Riemannian $g$-natural metric Kaluza-Klein type. Then considering class naturally defined almost complex structures on $TM$, constructed by V. Oproiu, construct magnetic fields and characterize corresponding curves when is space form.

Let (M, J) be a minimal compact complex surface of Kähler type. It is shown that the smooth 4-manifold M admits a Riemannian metric of positive scalar curvature iff (M,J) admits a Kähler metric of positive scalar curvature. This extends previous results of Witten and Kronheimer. A complex surface is a pair (M,J) consisting of a smooth compact 4-manifold M and a complex structure J on M ; the la...

Warped products provide a rich class of physically significant geometric objects. Warped product construction is an important method to produce a new metric with a base manifold and a fibre. We construct compact base manifolds with a positive scalar curvature which do not admit any non-trivial quasi-Einstein warped product, and non compact complete base manifolds which do not admit any non-triv...

Non blow-up of the 3D incompressible Euler Equations is proven for a class of threedimensional initial data characterized by uniformly large vorticity in bounded cylindrical domains. There are no conditional assumptions on the properties of solutions at later times, nor are the global solutions close to some 2D manifold. The approach of proving regularity is based on investigation of fast singu...

We consider foliations of complex projective manifolds by analytic curves. In a generic case each leaf is hyperbolic and there exists unique Poincaré metric on the leaves. It is shown that in a generic case this metric smoothly depends on a leaf. The manifold of universal covering of the leaves passing through some transversal base has a natural complex structure. It is shown that this structur...

Examples of almost-positively and quasi-positively curved spaces of the form M = H\((G, h) × F ) were discovered recently [9],[8]. Here h is a left-invariant metric on a compact Lie group G, F is a compact Riemannian manifold on which the subgroup H ⊂ G acts isometrically on the left, and M is the orbit space of the diagonal left action of H on (G, h)×F with the induced Riemannian submersion me...