Related papers: A Bonnet theorem for submanifolds into rotational …

We prove a Bonnet theorem for isometric immersions of submanifolds into the products of an arbitrary number of simply connected real space forms. Then, we prove the existence of associated families of minimal surfaces in such products.…

In this paper, we aim to provide a notion of "relative objects", i.e. objects equipped with some sort of subobjects, in differential topology. In spite of active researches relating them, e.g. knot theory or the theory of manifolds with…

Motivated by the theory of Inoue-type varieties, we give a structure theorem for projective manifolds $W_0$ with the property of admitting a 1-parameter deformation where $W_t$ is a hypersurface in a projective smooth manifold $Z_t$. Their…

We obtain a sub-Riemannian version of the classical Gauss-Bonnet theorem. We consider subsurfaces of a three dimensional contact sub-Riemannian manifolds, and using a family of taming Riemannian metric, we obtain a pure sub-Riemannian…

We prove a centre manifold theorem for a map along a manifold-with-boundary of fixed points, and provide an application to the study of gradient descent with large step size on two-layer matrix factorisation problems.

For compact Riemann surfaces, the collar theorem and Bers' partition theorem are major tools for working with simple closed geodesics. The main goal of this paper is to prove similar theorems for hyperbolic cone-surfaces. Hyperbolic…

Our goal is to give Schmidt's subspace theorem for moving hypersurface targets in subgeneral position in projective varieties.

We prove the Bonnet theorem for statistical manifolds, which states that if a statistical manifold admits tensors satisfying the Gauss--Codazzi--Ricci equations, then it is locally embeddable to a flat statistical manifold (or a Hessian…

On H-type sub-Riemannian manifolds we establish sub-Hessian and sub-Laplacian comparison theorems which are uniform for a family of approximating Riemannian metrics converging to the sub-Riemannian one. We also prove a sharp sub-Riemannian…

We establish the extrinsic Bonnet-Myers Theorem for compact Riemannian manifolds with positive Ricci curvature. And we show the almost rigidity for compact hypersurfaces, which have positive sectional curvature and almost maximal extrinsic…

In this paper, we study shape functions depending on closed submanifolds. We prove a new structure theorem that establishes the general structure of the shape derivative for this type of shape function. As a special case we obtain the…

A generic surface in Euclidean 3-space is determined uniquely by its metric and curvature. Classification of all special surfaces where this is not the case, i.e. of surfaces possessing isometries which preserve the mean curvature, is known…

A proof based on the Chern-Gauss-Bonnet Theorem is given to Hopf Theorem concerning the degree of the Gauss map of a hypersurface in $\mathbb{R}^n$.

This survey aims to provide a guide to the literature on topological 4-manifolds. Foundational theorems on 4-manifolds are stated, especially in the topological category. Precise references are given, with indications of the strategies…

In this paper, we apply techniques from equivariant geometry to prove that a generalized Bour's theorem holds for surfaces that are invariant under the action of a one-parameter group of isometries of a three-dimensional Riemannian…

In this paper, we establish a Schmidt's subspace theorem for moving hypersurfaces in weakly subgeneral position. Our result generalizes the previous results on Schmidt's theorem for the case of moving hypersurfaces.

We prove a Morse index theorem for action functionals on paths that are allowed to reflect at a hypersurface (either in the interior or at the boundary of a manifold). Both fixed and periodic boundary conditions are treated.

We extend the infinitesimal Torelli theorem for smooth hypersurfaces to nodal hypersurfaces.

The purpose of this article is twofold. The first is to prove a second main theorem for meromorphic mappings of $\C^m$ into a complex projective variety intersecting hypersurfaces in subgeneral position with truncated counting functions.…

In the present paper, we study timelike surfaces free of minimal points in the four-dimensional Minkowski space. For each such surface we introduce a geometrically determined pseudo-orthonormal frame field and writing the derivative…