arXiv research

Minimal normal curvature immersions in the unit ball studied.

We establish a nice orthonormal frame field on a closed surface minimally immersed in a unit sphere $S^{n}$, under which the shape operators take very simple forms. Using this frame field, we obtain an interesting property $K+K^{N}=1$ for the Gauss curvature $K$ and the normal curvature $K^{N}$ if the Gauss curvature i…

This note analyzes the normal form of gradient Ricci 4-solitons.

In this paper the geometry of normal metric contact pair manifolds is studied under the flatness of conformal, concircular and quasi-conformal curvature tensors. It is proved that a conformal flat normal metric contact pair manifold is an Einstein manifold with a negative scalar curvature and has positive sectional cur…

Classifies special submanifolds with specific curvature properties.

Veronese minimizes normal curvatures to sphere.

We prove that codimension two surfaces satisfying a nonlinear curvature condition depending on normal curvature are smoothly deformed by mean curvature flow to round points.

Lower bounds on average normal curvature for submanifolds in Riemannian domains.

Proves a special type of submanifolds in a curved space.

Study on generalized quasi-Einstein structures in contact geometry.

In this paper, we study normal complex contact metric manifolds and we get some general results on them. Moreover, we obtained the general expression of the curvature tensor field for arbitrary vector fields. Furthermore, we show that the necessary and succient conditions to be normal a complex contact metric manifold.…

In this paper, we proved the normal scalar curvature conjecture and the Bottcher-Wenzel conjecture.

In this paper we form relations for the determination of the elements of the Eötvös matrix of the Earth's normal gravity field. In addition a relation between the Gauss curvature of the normal equipotential surface and the Gauss curvature of the actual equipotential surface both passing through the point P is presented…

Study on immersions with flat normal bundle in curved spaces.

2-stein submanifolds in space forms have constant curvature if normal connection is flat or codimension is 2.

In this paper we consider Lorentzian surfaces in the 4-dimensional pseudo-Riemannian sphere $\mathbb S^4_2(1)$ with index 2 of curvature one. We obtain the complete classification of minimal Lorentzian surfaces $\mathbb S^4_2(1)$ whose Gaussian and normal curvatures are constants. We conclude that such surfaces have th…

Study on curves around a Whitney umbrella focusing on geodesic and normal curvatures.

We show that any normal metric on a closed biquotient with finite fundamental group has positive Ricci curvature.

Study examines preservation of curvature-adaptedness during mean curvature flow.

Optimal bounds found for torus curvatures in high dimensions.

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 …

Study on surfaces pinched by curvature in space forms converging under specific conditions.

New Ricci curvature means derived from plane curvatures.

Defines and analyzes generalized normal ruled surfaces of curves in 3D space.

The Ricci flow preserves positivity on Stiefel manifolds.

We obtain several rigidity results for biharmonic submanifolds in $\mathbb{S}^{n}$ with parallel normalized mean curvature vector field. We classify biharmonic submanifolds in $\mathbb{S}^{n}$ with parallel normalized mean curvature vector field and with at most two distinct principal curvatures. In particular, we dete…

We define cuspidal curvature $κ_c$ (resp. normalized cuspidal curvature $μ_c$) along cuspidal edges (resp. at swallowtail singularity) in Riemannian $3$-manifolds, and show that it gives a coefficient of the divergent term of the mean curvature function. Moreover, we show that the product $κ_Π$ called the product curva…

We consider graphs Sigma^n in R^m with prescribed mean curvature and flat normal bundle. Using techniques of Schoen, Simon and Yau, and Ecker-Huisken, we derive an interior curvature estimate of the form |A|^2<=C/R^2 up to dimension n<=5, where C is a constant depending on natural geometric data of Sigma^n only. This g…

Inverts operator on hyperbolic surfaces, constructing invariant distributions.

This paper deals with relative normalizations of skew ruled surfaces in the Euclidean space $\mathbb{E}^{3}$. In section 2 we investigate some new formulae concerning the Pick invariant, the relative curvature, the relative mean curvature and the curvature of the relative metric of a relatively normalized ruled surface…

In this paper, we proved the Normal Scalar Curvature Conjecture and the Bottcher-Wenzel Conjecture. We also established some new pinching theorems for minimal submanifolds in spheres.

The study preserves Ricci curvature positivity on homogeneous spaces under the normalized Ricci flow.

We construct a special class of Lorentz surfaces in the pseudo-Euclidean 4-space with neutral metric which are one-parameter systems of meridians of rotational hypersurfaces with timelike or spacelike axis and call them meridian surfaces. We give the complete classification of the meridian surfaces with parallel mean c…

The study shows how to foliate convex hypersurfaces in affine space with constant curvature.

We study surfaces with parallel normalized mean curvature vector field in Euclidean or Minkowski 4-space. On any such surface we introduce special isothermal parameters (canonical parameters) and describe these surfaces in terms of three invariant functions. We prove that any surface with parallel normalized mean curva…

We study a pointwise inequality for submanifolds in real space forms involving the scalar curvature, the normal scalar curvature and the mean curvature. We translate it into an algebraic problem, allowing us to prove a slightly weaker version of it. We also prove the conjecture for certain types of submanifolds of $\ma…

The paper characterizes surfaces in 4D space forms with flat normal connection.

New inequalities for austere submanifolds established.

Proves spheres with bounded curvatures must contain a unit ball.

Let M be a compact pseudo-umbilical submanifold of the unit sphere S. In the present note, it is shown that if the normal curvature, scalar curvature S and square of the length of second fundamental form satisfy certain conditions, then M is totally geodesic.

We discuss some consequences of the existence of the holomorphic quadratic Hopf differential on a conformally immersed constant mean curvature topological disc with analytic boundary. In particular, we derive a formula for the mean curvature as a weighted average of the normal curvature of the boundary curve, and a con…

This paper studies mean curvature flows near cylindrical singularities.

The paper studies special surfaces in 4D space forms with specific geometric properties.

In this paper we obtain two types of optimal inequalities consisting of the normalized scalar curvature and the generalized normalized $δ$-Casorati curvatures for real hypersurfaces of complex two-plane Grassmannians and complex hyperbolic two-plane Grassmannians. We also find the conditions on which the equalities hol…

The study examines stability of triharmonic hypersurfaces in space forms.

The paper defines and studies new types of submanifolds in a unit sphere.

Revisits conformal metrics with finite Q-curvature, providing necessary and sufficient conditions.

The paper resolves a conjecture about curvature conditions on manifolds.