arXiv research

The study proves that pseudo-umbilical spacelike submanifolds are totally geodesic and totally umbilical.

Invariant counts maximum stable umbilic splits.

The paper studies umbilics on surfaces in Lorentz-Minkowski space.

Defines conditions for umbilical submanifolds in arbitrary dimensions.

Minimal totally real submanifolds in complex space forms have special umbilical properties.

The paper finds conditions for certain hypersurfaces to be totally umbilical.

The study proves umbilical points have positive measure in 3D space forms.

For a regular surface in Euclidean space $\mathbb{R}^3$, umbilic points are precisely the points where the Gauss and mean curvatures $K$ and $H$ satisfy $H^2=K$; moreover, it is well-known that the only totally umbilic surfaces in $\mathbb{R}^3$ are planes and spheres. But for timelike surfaces in Minkowski space $\mat…

Paper characterizes umbilical hypersurfaces using a generalized overdetermined problem.

Characterizes curves in totally umbilical surfaces of space forms.

The study defines conditions for mean curvature of umbilical leaves in hyperbolic space.

The paper studies quasi-umbilical timelike surfaces in a specific geometric setting.

Study on null hypersurfaces in complex contact manifolds.

Paper solves a mixed boundary value problem in space forms with umbilical boundaries.

Study finds unique totally umbilic null hypersurfaces in specific spacetimes.

The study characterizes submanifolds in product spaces.

The study classifies totally umbilical surfaces in warped product manifolds.

Characterizes totally umbilical hypersurfaces in product spaces.

Willmore surfaces have umbilic points forming a smooth manifold.

The paper classifies submanifolds in pseudo-Riemannian space forms.

New geometric proof shows index of umbilic points on analytic surfaces is at most one.

Classifies special submanifolds in geometric spaces.

In [2] we have classified the Blaschke quasi-umbilical submanifolds in the conformal space ${\mathbb Q}^n_s$. In this paper we shall classify the Blaschke para-umbilical hypersurfaces in the conformal space ${\mathbb Q}^n_s$. That may be also considered as the extension of the classification of the conformal isotropic …

Study on real hypersurfaces in products of complex space forms, proving rigidity and nonexistence results.

Investigates totally umbilical immersions in R^3 with special metrics.

Paper generalizes Minkowski inequality for umbilical hypersurfaces with free boundary.

The paper studies minimal submanifolds with specific curvature properties in Euclidean space.

The authors obtained the classification theorem for Lagrangian H-umbilical submanifolds in quaternion Euclidean spaces using the idea of quaternion extensors.

Study of $f$-biharmonic hypersurfaces in conformally flat spaces.

New characterizations of umbilic hypersurfaces in warped product manifolds.

Using the idea of special Legendre curves, the authors obtained the explicit description of flat Lagrangian H-umbilical submanifolds in quaternion Euclidean spaces.

A new definition of umbilic points at infinity for polynomial surfaces.

Study of umbilic points on Willmore surfaces in 3-sphere.

The paper studies biharmonic conformal hypersurfaces in Riemannian manifolds.

A spacelike surface S immersed in a 4-dimensional Lorentzian manifold will be said to be umbilical along a direction N normal to S if the second fundamental form along N is proportional to the first fundamental form of S. In particular, S is pseudo-umbilical if it is umbilical along the mean curvature vector field H, a…

New metrics contradicting old conjectures on umbilic points.

Study on umbilical submanifolds in specific geometric spaces.

The notion of Lagrangian $H$-umbilical submanifolds was introduced by B. Y. Chen in 1997, and these submanifolds have appeared in several important problems in the study of Lagrangian submanifolds from the Riemannian geometric point of view. Recently, the author introduced the notion of tangentially biharmonic submanif…

We give a complete classification of umbilical surfaces of arbitrary codimension of a product $Q^{n_1}_{k_1}\times Q^{n_2}_{k_2}$ of space forms whose curvatures satisfy $k_1 + k_2 \not= 0$.

The paper describes the CR umbilical locus of a real ellipsoid in complex space.

We find explicitly all bi-umbilical foliated semi-symmetric hypersurfaces in the four-dimensional Euclidean space.

Study affine curvature lines on surfaces in 3D space.

Study classifies submanifolds in probability simplex.

In this article, we prove new stability results for almost-Einstein hypersurfaces of the Euclidean space, based on previous eigenvalue pinching results. Then, we deduce some comparable results for almost umbilical hypersurfaces.

Several characterizations of umbilic points of submanifolds in arbitrary Riemannian and Lorentzian manifolds are given. As a consequence, we obtain new characterizations of spheres in the Euclidean space and of hyperbolic spaces in the Lorentz-Minkowski space. We also prove the Lorentzian version of a classical result …

Study on stability of mean curvature flow in hyperbolic space.

Solutions grow for a special type of math problem on curved spaces.

Following ideas of Choe and Fernandez-do Carmo, we give sufficient conditions for a disk type surface, with piecewise smooth boundary, to be totally umbilical for a given Coddazi pair. As a consequence, we obtain rigidity results for surfaces in space forms and in homogeneous product spaces that generalizes some known …