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...

We establish a new algebraic characterization of sectional curvature bounds $\sec\geq k$ and $\sec\leq using only terms in the Weitzenb\"ock formulae for symmetric $p$-tensors. By introducing analogue Kulkarni-Nomizu product, we provide simple formula such terms. also give an application Bochner technique to closed $4$-manifolds with indefinite intersection form $\sec>0$ or $\sec\geq0$, obtaini...

Abstract The in situ and real-time measurement of curvature changes optically reflecting surfaces is a key element to better control bottom-up fabrication processes the semiconductor industry, but also follow or adjust mirror deformations during use for space optics industries. Despite progresses made last two decades thanks laser deflectometry-based techniques, community lacks an instrument, e...

We examine the class of compact Hermitian manifolds with constant holomorphic sectional curvature. Such are conjectured to be K\"ahler (hence a complex space form) when is non-zero and Chern flat quotient Lie group) zero. The conjecture known in dimension two but open higher dimensions. In this paper, we establish partial solution three by proving that any threefold zero real bisectional curvat...

Let M be a compact (two-sided) minimal hypersurface in a Riemannian manifold M n+1 . It is a simple fact that if M has positive Ricci curvature then M cannot be stable (i. e. its Jacobi operator L has index at least one). If M = S(1) is the unit sphere and L has index one, then it is known that M must be a totally geodesic equator. We prove that if M is the real projective space RP = Sn+1(1)/{±...

We study the semi-Riemannian geometry of foliation F an indefinite locally conformal Kähler (l.c.K.) manifold M, given by Pfaffian equation ω=0, provided that ∇ω=0 and c=∥ω∥≠0 (ω is Lee form M). If M conformally flat then every leaf shown to be a totally geodesic hypersurface in space sectional curvature c/4, carrying c-Sasakian structure. As corollary result together with version de Rham decom...

We characterize the natural diagonal almost product (locally product) structures on the tangent bundle of a Riemannian manifold. We obtain the conditions under which the tangent bundle endowed with the determined structure and with a metric of natural diagonal lift type is a Riemannian almost product (locally product) manifold, or an (almost) para-Hermitian manifold. We find the natural diagona...