No Arabic abstract
In a Riemannian manifold, the existence of a new connection is proved. In particular cases, this connection reduces to several symmetric, semi-symmetric and quarter-symmetric connections; even some of them are not introduced so far. We also find formula for curvature tensor of this new connection.
We define kinetic Brownian motion on the diffeomorphism group of a closed Riemannian manifold, and prove that it provides an interpolation between the hydrodynamic flow of a fluid and a Brownian-like flow.
Let $M^n$ be a closed convex hypersurface lying in a convex ball $B(p,R)$ of the ambient $(n+1)$-manifold $N^{n+1}$. We prove that, by pinching Heintze-Reillys inequality via sectional curvature upper bound of $B(p,R)$, 1st eigenvalue and mean curvature of $M$, not only $M$ is Hausdorff close and almost isometric to a geodesic sphere $S(p_0,R_0)$ in $N$, but also its enclosed domain is $C^{1,alpha}$-close to a geodesic ball of constant curvature.
In this paper, we give an explicit second variation formula for a biharmonic hypersurface in a Riamannian manifold similar to that of a minimal hypersurface. We then use the second variation formula to compute the stability index of the known biharmonic hypersurfaces in a Euclidean sphere, and to prove the non-existence of unstable proper biharmonic hypersurface in a Euclidean space or a hyperbolic space, which adds another special case to support Chens conjecture on biharmonic submanifolds.
We give a differential geometric construction of a connection in the bundle of quantum Hilbert spaces arising from half-form corrected geometric quantization of a prequantizable, symplectic manifold, endowed with a rigid, family of Kahler structures, all of which give vanishing first Dolbeault cohomology groups. In [And1] Andersen gave an explicit construction of Hitchins connection in the non-corrected case using additional assumptions. Under the same assumptions we also give an explicit solution in terms of Ricci potentials. Morover we show that if these are carefully chosen the construction coincides with the construction of Andersen in the non-corrected case.
Let $M$ be an $n$-dimensional complete simply connected Riemannian manifold with sectional curvature bounded above by a nonpositive constant $-kappa^2$. Using the cone total curvature $TC(Gamma)$ of a graph $Gamma$ which was introduced by Gulliver and Yamada Math. Z. 2006, we prove that the density at any point of a soap film-like surface $Sigma$ spanning a graph $Gamma subset M$ is less than or equal to $frac{1}{2pi}{TC(Gamma) - kappa^{2}area(pmbox{$timeshspace*{-0.178cm}times$}Gamma)}$. From this density estimate we obtain the regularity theorems for soap film-like surfaces spanning graphs with small total curvature. In particular, when $n=3$, this density estimate implies that if begin{eqnarray*} TC(Gamma) < 3.649pi + kappa^2 inf_{pin M} area({pmbox{$timeshspace*{-0.178cm}times$}Gamma}), end{eqnarray*} then the only possible singularities of a piecewise smooth $(mathbf{M},0,delta)$-minimizing set $Sigma$ is the $Y$-singularity cone. In a manifold with sectional curvature bounded above by $b^2$ and diameter bounded by $pi/b$, we obtain similar results for any soap film-like surfaces spanning a graph with the corresponding bound on cone total curvature.