One of the most important properties of a geometric flow is whether it preserves the positivity of various notions of curvature. In the case of the Kähler-Ricci flow, the positivity of the curvature operator (Hamilton [7]), the positivity of the biholomorphic sectional curvature (Bando [1], Mok[8]), and the positivity of the scalar curvature (Hamilton [4]) are all preserved. However, whether th...

In this paper, we study half-lightlike submanifolds of a semi-Riemannian manifold such that the shape operator of screen distribution is conformal to the shape operator of screen transversal distribution. We mainly obtain some results concerning the induced Ricci curvature tensor and the null sectional curvature of screen transversal conformal half-lightlike submanifolds. 2010 Mathematics Subje...

i ii Chapter 1 Introduction In [11], Hamilton determined a sharp differential Harnack inequality of Li–Yau type for complete solutions of the Ricci flow with non-negative curvature operator. This Li–Yau–Hamilton inequality (abbreviated as LYH inequality below) is of critical importance to the understanding of singularities of the Ricci flow, as is evident from its numerous applications in [10],...

We exhibit several families of Jacobi–Videv pseudo-Riemannian manifolds which are not Einstein. We also exhibit Jacobi–Videv algebraic curvature tensors where the Ricci operator defines an almost complex structure.

Consider {(M, g(t)), 0 ≤ t < T < ∞} as an unnormalized Ricci flow solution: dgij dt = −2Rij for t ∈ [0, T ). Richard Hamilton shows that if the curvature operator is uniformly bounded under the flow for all t ∈ [0, T ) then the solution can be extended over T . Natasa Sesum proves that a uniform bound of Ricci tensor is enough to extend the flow. We show that if Ricci is bounded from below, the...

Spectrum of kinematic fast dynamo operators in Ricci compressible flows in Einstein 2-manifolds is investigated. A similar expression, to the one obtained by Chicone, Latushkin and Montgomery-Smith (Comm Math Phys (1995)) is given, for the fast dynamo operator. The operator eigenvalue is obtained in a highly conducting media, in terms of linear and nonlinear orders of Ricci scalar and diffusion...

In [H4], Hamilton determined a sharp tensor Harnack inequality for complete solutions of the Ricci flow with non-negative curvature operator. This Harnack inequality is of critical importance to the understanding of singularities of the Ricci flow, as is evident from its numerous applications in [H3], [H5], [H6], and [H7]. Moreover, according to Hamilton, the discovery of a Harnack inequality i...

Perelman [Pe02] has discovered a remarkable variational structure for the Ricci flow: it can be viewed as the gradient flow of the entropy functional λ. There are also two monotonicity formulas of shrinking or localizing type: the shrinking entropy ν, and the reduced volume. Either of these can be seen as the analogue of Huisken’s monotonicity formula for mean curvature flow [Hu90]. In various ...

The famous Frankel conjecture asserts that any compact Kähler manifold with positive bisectional curvature must be biholomorphic to CP n. This conjecture was settled affirmatively in early 1980s by two groups of mathematicians independently: Siu-Yau[16] via differential geometry method and Morri [15] by algebraic method. There are many interesting papers following this celebrated work; in parti...

We derive matter collineations for some static spherically symmetric spacetimes and compare the results with Killing, Ricci and Curvature symmetries. We conclude that matter and Ricci collineations are not, in general, the same.