In this paper, we show that an $L_1$-2-type surface in the three-dimensional hyperbolic space $H^3subset R^4_1$ either is an open piece of a standard Riemannian product $ H^1(-sqrt{1+r^2})times S^{1}(r)$, or it has non constant mean curvature, non constant Gaussian curvature, and non constant principal curvatures.

which is sharp as indicated in the Euclidean case. However even if M contains a small compact region where the Ricci curvature is not nonnegative, estimate (1.1) becomes very much different from (1.2) when r is large, due to the presence of the √ k term. Whether estimate (1.2) is stable under perturbation has been an open question for some time, in light of the known stability results on weaker...

The combinatorial Ricci curvature of Forman, which is defined at the edges of a CW complex, and which makes use of only the face relations of the cells in the complex, does not satisfy an analog of the Gauss-Bonnet Theorem, and does not behave analogously to smooth surfaces with respect to negative curvature. We extend this curvature to vertices and faces in such a way that the problems with co...

Combining several previously known arguments, we prove marked length spectrum rigidity for surfaces with nonpositively curved Riemannian metrics away from a finite set of cone-type singularities with cone angles > 2π. With an additional condition, we can weaken the requirement on one metric to ‘no conjugate points.’

In this paper we prove two sharp inequalities involving the normalized scalar curvature and the generalized normalized δ-Casorati curvatures for slant submanifolds in quaternionic space forms. We also characterize those submanifolds for which the equality cases hold. These results are a generalization of some recent results concerning the Casorati curvature for a slant submanifold in a quaterni...

where ω̃ = ( √ −1/2)g̃ij̄dz ∧ dz and Σ̃ = ( √ −1/2)R̃ij̄dz ∧ dz are the Kähler form, the Ricci form of the metric g̃ respectively, while c1(M) denotes the first Chern class. Under the normalized initial condition (2), the first author [3] (see also Proposition 1.1 in [4]) showed that the solution g(x, t) = ∑ gij̄(x, t)dz dz to the normalized flow (1) exists for all time. Furthermore by the work of Mok ...

One of the basic problems of Riemannian geometry is the classification of manifolds of positive sectional curvature. The known examples include the spherical space forms which carry constant curvature metrics and the rank 1 symmetric spaces whose canonical metrics have sectional curvatures at each point varying between 1 and 4. In 1951 H.E. Rauch [18] introduced the notion of curvature pinching...

Let M be a compact k-dimensional riemmanian manifold minimally immersed in the unit n-dimensional sphere S. It is easy to show that for any p ∈ S the boundary of the geodesic ball in S with radius π2 and center at p (in this case this boundary is an equator) must intercept the manifold M . When the codimension is 1, i.e. k = n − 1, it is known that the ricci curvature, is not greater than 1. We...

In this article, we will give a brief survey on some recent development concerning the understanding of the structure at infinity of a complete manifold whose spectrum has a positive lower bound. Throughout this paper, we denote M to be a complete n-dimensional manifold without boundary endowed with the metric ds. We assume that the Ricci curvature of M is bounded from below by some constant. R...

The conformal monogenic signal is a novel rotational invariant approach for analyzing i(ntrinsic)1D and i2D local features of twodimensional signals (e.g. images) without the use of any heuristics. It contains the monogenic signal as a special case for i1D signals and combines scale-space, phase, orientation, energy and isophote curvature in one unified algebraic framework. The conformal monoge...