Real hypersurfaces in the complex hyperbolic quadric with normal Jacobi operator of Codazzi type

In this paper, we introduce the notion of normal Jacobi operator of Codazzi type for real hypersurfaces in the complex hyperbolic quadric [Formula: see text]. The normal Jacobi operator of Codazzi type implies that the unit normal vector field [Formula: see text] becomes [Formula: see text]-principal or [Formula: see text]-isotropic. Then according to each case, we give a complete classification of Hopf real hypersurfaces in [Formula: see text] with normal Jacobi operator of Codazzi type. The result of the classification shows that no such hypersurfaces exist.

L p -trace-free generalized Korn inequalities for incompatible tensor fields in three space dimensions

For $1< p<\infty$ we prove an $L^{p}$ -version of the generalized trace-free Korn inequality for incompatible tensor fields $P$ in $W^{1,p}_0(\operatorname {Curl}; \Omega ,\mathbb {R}^{3\times 3})$ . More precisely, let $\Omega \subset \mathbb {R}^{3}$ be a bounded Lipschitz domain. Then there exists a constant $c>0$ such that \[ \lVert{ P }\rVert_{L^{p}(\Omega,\mathbb{R}^{3\times 3})}\leq c\,\left(\lVert{\operatorname{dev} \operatorname{sym} P }\rVert_{L^{p}(\Omega,\mathbb{R}^{3\times 3})} + \lVert{ \operatorname{dev} \operatorname{Curl} P }\rVert_{L^{p}(\Omega,\mathbb{R}^{3\times 3})}\right) \] holds for all tensor fields $P\in W^{1,p}_0(\operatorname {Curl}; \Omega ,\mathbb {R}^{3\times 3})$ , i.e., for all $P\in W^{1,p} (\operatorname {Curl}; \Omega ,\mathbb {R}^{3\times 3})$ with vanishing tangential trace $P\times \nu =0$ on $\partial \Omega$ where $\nu$ denotes the outward unit normal vector field to $\partial \Omega$ and $\operatorname {dev} P : = P -\frac 13 \operatorname {tr}(P) {\cdot } {\mathbb {1}}$ denotes the deviatoric (trace-free) part of $P$ . We also show the norm equivalence \begin{align*} &\lVert{ P }\rVert_{L^{p}(\Omega,\mathbb{R}^{3\times 3})}+\lVert{ \operatorname{Curl} P }\rVert_{L^{p}(\Omega,\mathbb{R}^{3\times 3})}\\ &\quad\leq c\,\left(\lVert{P}\rVert_{L^{p}(\Omega,\mathbb{R}^{3\times 3})} + \lVert{ \operatorname{dev} \operatorname{Curl} P }\rVert_{L^{p}(\Omega,\mathbb{R}^{3\times 3})}\right) \end{align*} for tensor fields $P\in W^{1,p}(\operatorname {Curl}; \Omega ,\mathbb {R}^{3\times 3})$ . These estimates also hold true for tensor fields with vanishing tangential trace only on a relatively open (non-empty) subset $\Gamma \subseteq \partial \Omega$ of the boundary.

On f-rectifying curves in the Euclidean 4-space

A rectifying curve in the Euclidean 4-space 𝔼4 is defined as an arc length parametrized curve γ in 𝔼4 such that its position vector always lies in its rectifying space (i.e., the orthogonal complement Nγ ˔ of its principal normal vector field Nγ) in 𝔼4. In this paper, we introduce the notion of an f-rectifying curve in 𝔼4 as a curve γ in 𝔼4 parametrized by its arc length s such that its f-position vector γf, defined by γf (s) = ∫ f(s)dγ for all s, always lies in its rectifying space in 𝔼4, where f is a nowhere vanishing integrable function in parameter s of the curve γ. Also, we characterize and classify such curves in 𝔼4.

Integral Formulas for a Foliation with a Unit Normal Vector Field

In this article, we prove integral formulas for a Riemannian manifold equipped with a foliation F and a unit vector field N orthogonal to F, and generalize known integral formulas (due to Brito-Langevin-Rosenberg and Andrzejewski-Walczak) for foliations of codimension one. Our integral formulas involve Newton transformations of the shape operator of F with respect to N and the curvature tensor of the induced connection on the distribution D=TF⊕span(N), and this decomposition of D can be regarded as a codimension-one foliation of a sub-Riemannian manifold. We apply our formulas to foliated (sub-)Riemannian manifolds with restrictions on the curvature and extrinsic geometry of the foliation.

Hypersurfaces of a Sasakian manifold - revisited

We study orientable hypersurfaces in a Sasakian manifold. The structure vector field ξ of a Sasakian manifold determines a vector field v on a hypersurface that is the component of the Reeb vector field ξ tangential to the hypersurface, and it also gives rise to a smooth function σ on the hypersurface, namely the projection of ξ on the unit normal vector field N. Moreover, we have a second vector field tangent to the hypersurface, given by $\mathbf{u}=-\varphi (N)$ u = − φ ( N ) . In this paper, we first find a necessary and sufficient condition for a compact orientable hypersurface to be totally umbilical. Then, with the assumption that the vector field u is an eigenvector of the Laplace operator, we find a necessary condition for a compact orientable hypersurface to be isometric to a sphere. It is shown that the converse of this result holds, provided that the Sasakian manifold is the odd dimensional sphere $\mathbf{S}^{2n+1}$ S 2 n + 1 . Similar results are obtained for the vector field v under the hypothesis that this is an eigenvector of the Laplace operator. Also, we use a bound on the integral of the Ricci curvature $Ric ( \mathbf{u},\mathbf{u} ) $ R i c ( u , u ) of the compact hypersurface to find a necessary condition for the hypersurface to be isometric to a sphere. We show that this condition is also sufficient if the Sasakian manifold is $\mathbf{S}^{2n+1}$ S 2 n + 1 .

A new classification on parallel Ricci tensor for real hypersurfaces in the complex quadric

First we introduce the notion of parallel Ricci tensor ${\nabla }\mathrm {Ric}=0$ for real hypersurfaces in the complex quadric Q m = SOm+2/SO m SO2 and show that the unit normal vector field N is singular. Next we give a new classification of real hypersurfaces in the complex quadric Q m = SOm+2/SO m SO2 with parallel Ricci tensor.

Discrete Normal Vector Field Approximation via Time Scale Calculus

The theory of time scales calculus have long been a subject to many researchers from different disciplines. Beside the unification and the extension aspects of the theory, it emerge as a powerful tool for mimetic discretization process. In this study, we present a framework to find normal vector fields of discrete point sets in ℝ3 by using symmetric differential on time scales. A surface parameterized by the tensor product of two time scales can be analogously expressed as the vertex set of non-regular rectangular grids. If the time scales are dense, then the discrete grid structure vanishes. If the time scales are isolated, then the further geometric analysis can be executed by using symmetric dynamic differential. Moreover, we present an algorithmic procedure to determine the symmetric dynamic differential structure on the neighborhood of points in surfaces. Our results indicate that the method we present has good approximation to unit normal vector fields of parameterized surfaces rather than the Delaunay triangulation for some points.