One of the most fundamental problems in submanifold theory is to establish simple relationships between intrinsic and extrinsic invariants of the submanifolds (cf. [6]). A general optimal inequality for submanifolds in Riemannian manifolds of constant sectional curvature was obtained in an earlier article [5]. In this article we extend this inequality to a general optimal inequality for arbitra...

In this paper we introduce a new symmetrization with respect to mixed volume or anisotropic curvature integral, which generalizes the one quermassintegral due Talenti [26] and Tso [28]. We show Pólya-Szegő type principle for such — it diminishes Hessian integral quasi-convex functions. achieve by systematic study of invariants on non-symmetric matrices real eigenvalues higher order mean curvatu...

We show that a noncompact manifold with bounded sectional curvature, whose ends are sufficiently collapsed, has a finite dimensional space of square-integrable harmonic forms. In the special case of a finite-volume manifold with pinched negative sectional curvature, we show that the essential spectrum of the p-form Laplacian is the union of the essential spectra of a collection of ordinary diff...

The flag curvature is a natural extension of the sectional curvature in Riemannian geometry, and the S-curvature is a non-Riemannian quantity which vanishes for Riemannian metrics. There are (incomplete) nonRiemannian Finsler metrics on an open subset in Rn with negative flag curvature and constant S-curvature. In this paper, we are going to show a global rigidity theorem that every Finsler met...

In Kähler geometry, Fujiki–Donaldson show that the scalar curvature arises as moment map for Hamiltonian diffeomorphisms. generalized one does not have good notions of Levi-Civita connection and curvature, however there still exists a precise framework is defined (Goto, J Sympl Geom 18 (1):147–190, 2020). Then fundamental question to understand existence or non-existence structures with constan...

We show that the unit tangent bundle of S4 and a real cohomology CP 3 admit Riemannian metrics with positive sectional curvature almost everywhere. These are the only examples so far with positive curvature almost everywhere that are not also known to admit positive curvature. AMS Classi cation numbers Primary: 53C20 Secondary: 53C20, 58B20, 58G30