We describe all almost contact metric, almost hermitian and G 2-structures admitting a connection with totally skew-symmetric torsion tensor, and prove that there exists at most one such connection. We investigate its torsion form, its Ricci tensor, the Dirac operator and the ∇-parallel spinors. In particular, we obtain solutions of the type II string equations in dimension n = 5, 6 and 7.

In this paper, we present a new proof of the upper and lower bound estimates for the first Dirichlet eigenvalue λ1 (B (p, r)) of Laplacian operator for the manifold with Ricci curvature Rc ≥ −K, by using Li-Yau’s gradient estimate for the heat equation.

Abstract. We study the convergence of the Kähler-Ricci flow on a Fano manifold under some stability conditions. More precisely we assume that the first eingenvalue of the ∂̄-operator acting on vector fields is uniformly bounded along the flow, and in addition the Mabuchi energy decays at most logarithmically. We then give different situations in which the condition on the Mabuchi energy holds.

We study the dispersive properties of the wave equation associated with the shifted Laplace–Beltrami operator on Damek–Ricci spaces, and deduce Strichartz estimates for a large family of admissible pairs. As an application, we obtain global well–posedness results for the nonlinear wave equation.

In this note we introduce and study dynamical systems related to the Ricci operator on the space of Kähler metrics as discretizations of certain geometric flows. We pose a conjecture on their convergence towards canonical Kähler metrics and provide a proof in the case the first Chern class is negative, zero or positive.

We determine the matrix of the balanced metric of the Siegel–Jacobi ball and its inverse. We calculate the scalar curvature, the Ricci form and the Laplace–Beltrami operator of this manifold. We discuss several geometric aspects related with Berezin quantization on the Siegel–Jacobi ball.

For compact Kählerian manifolds, the holomorphic pseudosymmetry reduces to the local symmetry if additionally the scalar curvature is constant and the structure function is non-negative. Similarly, the holomorphic Ricci-pseudosymmetry reduces to the Ricci-symmetry under these additional assumptions. We construct examples of non-compact essentially holomorphically pseudosymmetric Kählerian manif...

An invariant cone in the space of curvature operators is one that is preserved by a flow. For Ricci flow, the condition R ≥ 0 is preserved in all dimensions, while the conditionR ≤ 0 is preserved only in real dimension two. Positive curvature operator is preserved in all dimensions [11], but positive sectional curvature is not preserved in dimensions four and above. The known counterexamples, c...