second fundamental form
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In this study, a link between the squared norm of the second fundamental form and the Laplacian of the warping function for a warped product pointwise semi-slant submanifold Mn in a complex projective space is presented. Some characterizations of the base NT of Mn are offered as applications. We also look at whether the base NT is isometric to the Euclidean space Rp or the Euclidean sphere Sp, subject to some constraints on the second fundamental form and warping function.

In this article, we obtain certain bounds for statistical curvatures of submanifolds with any codimension of Kenmotsu-like statistical manifolds. In this context, we construct a class of optimum inequalities for submanifolds in Kenmotsu-like statistical manifolds containing the normalized scalar curvature and the generalized normalized Casorati curvatures. We also define the second fundamental form of those submanifolds that satisfy the equality condition. On Legendrian submanifolds of Kenmotsu-like statistical manifolds, we discuss a conjecture for Wintgen inequality. At the end, some immediate geometric consequences are stated.

This paper studies the contact CR-warped product submanifolds of a generalized Sasakian space form admitting a nearly cosymplectic structure. Some inequalities for the existence of these types of warped product submanifolds are established, the obtained inequalities generalize the results that have acquired in \cite{atceken14}. Moreover, we also estimate another inequality for the second fundamental form in the expressions of the warping function, this inequality also generalizes the inequalities that have obtained in \cite{ghefari19}. In addition, we also explore the equality cases.

Abstract In this paper, we prove that a two-dimensional self-shrinker, homeomorphic to the sphere, immersed in the three-dimensional Euclidean space ℝ 3 {{\mathbb{R}}^{3}} is a round sphere, provided its mean curvature and the norm of the its position vector have an upper bound in terms of the norm of its traceless second fundamental form. The example constructed by Drugan justifies that the hypothesis on the second fundamental form is necessary. We can also prove the same kind of rigidity results for surfaces with parallel weighted mean curvature vector in ℝ n {{\mathbb{R}}^{n}} with radial weight. These results are applications of a new generalization of Cauchy’s Theorem in complex analysis which concludes that a complex function is identically zero or its zeroes are isolated if it satisfies some weak holomorphy.

In this paper, we obtain the classification theorems for 3-dimensional complete [Formula: see text]-translators [Formula: see text] with constant squared norm [Formula: see text] of the second fundamental form and constant [Formula: see text] in the Euclidean space [Formula: see text].

Locally strictly convex surfaces in four-dimensional affine space are studied from a perspective of the affine structure invented by Nuño-Ballesteros and Sánchez, which is especially suitable in convex geometry. The surfaces that are embedded in locally strictly convex hyperquadrics are classified under assumptions that the second fundamental form is parallel with respect to the induced connection and the normal connection is compatible with a metric on the transversal bundle. Both connections are induced by a canonical transversal plane bundle, which is defined by certain symmetry conditions. The obtained surfaces are always products of an ellipse and a conical planar curve.

The biwarped product submanifolds generalize the class of product submanifolds and are particular case of multiply warped product submanifolds. The present paper studies the biwarped product submanifolds of the type S T × ψ 1 S ⊥ × ψ 2 S θ in Sasakian space forms S ¯ c , where S T , S ⊥ , and S θ are the invariant, anti-invariant, and pointwise slant submanifolds of S ¯ c . Some characterizing inequalities for the existence of such type of submanifolds are proved; besides these inequalities, we also estimated the norm of the second fundamental form.

Abstract We prove that, for every closed (not necessarily convex) hypersurface Σ in ℝ n + 1 {\mathbb{R}^{n+1}} and every p > n {p>n} , the L p {L^{p}} -norm of the trace-free part of the anisotropic second fundamental form controls from above the W 2 , p {W^{2,p}} -closeness of Σ to the Wulff shape. In the isotropic setting, we provide a simpler proof. This result is sharp since in the subcritical regime p ≤ n {p\leq n} , the lack of convexity assumptions may lead in general to bubbling phenomena. Moreover, we obtain a stability theorem for quasi-Einstein (not necessarily convex) hypersurfaces and we improve the quantitative estimates in the convex setting.

In the present paper, by considering the Gauss equation in place of the Codazzi equation, we derive new optimal inequality for the second fundamental form of CR-warped product submanifolds into a generalized Sasakian space form. Moreover, the inequality generalizes some inequalities for various ambient space forms.