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Valery Marenich

Publications and source records attributed to Valery Marenich.

6 recordsLinked to original sources
arXiv2005

On non-negatively curved metrics on open five-dimensional manifolds

Let $V^n$ be an open manifold of non-negative sectional curvature with a soul $Σ$ of co-dimension two. The universal cover $\tilde N$ of the unit normal bundle $N$ of the soul in such a manifold is isometric to the direct product $M^{n-2}\times R$. In the study of the metric structure of $V^n$ an important role plays the vector field $X$ which belongs to the projection of the vertical planes distribution of the Riemannian submersion $π:V\toΣ$ on the factor $M$ in this metric splitting $\tilde N=M\times R$. The case $n=4$ was considered in [GT] where the authors prove that $X$ is a Killing vector field while the manifold $V^4$ is isometric to the quotient of $M^2\times (R^2,g_F)\times R$ by the flow along the corresponding Killing field. Following an approach of [GT] we consider the next case $n=5$ and obtain the same result under the assumption that the set of zeros of $X$ is not empty. Under this assumption we prove that both $M^3$ and $Σ^3$ admit an open-book decomposition with a bending which is a closed geodesic and pages which are totally geodesic two-spheres, the vector field $X$ is Killing, while the whole manifold $V^5$ is isometric to the quotient of $M^3\times (R^2,g_F)\times R$ by the flow along corresponding Killing field.

math.DG↗
arXiv2004

Rigidity of non-negatively curved metrics on open five-dimensional manifolds

As the first step in the direction of the Hopf conjecture on the non-existence of metrics with positive sectional curvature on $S^2 \times S^2$ D.Gromoll and K.Tapp in [GT] suggested the following (Weak Hopf) conjecture (on the rigidity of non-negatively curved metrics on $S^2 \times R^3$): "The boundary $S^2\times S^2$ of the $S^2 \times B^3\subset S^2 \times R^3$ with an arbitrary complete metric of non-negative sectional curvature contains a point where a curvature of $S^2 \times S^2$ vanish". In this note we verify this.

math.DG↗
arXiv2003

On Singular Connections and Geometrically Atomic Maps

We prove that the introduction of the class of geometrically atomic bundle maps by Harvey and Lawson in their theory of singular connections is not necessary because an arbitrary map satisfies the conditions of geometric atomicity.

math.DG↗