Real Hypersurfaces Research Articles

Smooth connectivity in real algebraic varieties

AbstractA standard question in real algebraic geometry is to compute the number of connected components of a real algebraic variety in affine space. This manuscript provides algorithms for computing the number of connected components, the Euler characteristic, and deciding the connectivity between two points for a smooth manifold arising as the complement of a real hypersurface of a real algebraic variety. When considering the complement of the set of singular points of a real algebraic variety, this yields an approach for determining smooth connectivity in a real algebraic variety. The method is based upon gradient ascent/descent paths on the real algebraic variety inspired by a method proposed by Hong, Rohal, Safey El Din, and Schost for complements of real hypersurfaces. Several examples are included to demonstrate the approach.

$$\mathcal {D}$$-Parallel structure Lie operators on real hypersurfaces in nonflat complex space forms

$$\mathcal {D}$$-Parallel structure Lie operators on real hypersurfaces in nonflat complex space forms

Chen-Type Inequality for Generic Submanifolds of Quaternionic Space Form and Its Application

In 1993, the theory of Chen invariants started when Chen wrote basic inequalities for submanifolds in space forms. This inequality is called Chen’s first inequality. Afterward, many geometers studied many papers dealing with this new inequality. The present paper aims to establish a Chen inequality for quaternionic generic submanifolds in a quaternionic space form and obtain this inequality for real hypersurfaces.

Purity and hybridness of two tensors on a real hypersurface in complex projective space

Abstract On a real hypersurface M M in complex projective space, we can define two tensor fields of type (1, 2), A F ( k ) {A}_{F}^{\left(k)} and A T ( k ) {A}_{T}^{\left(k)} , associated with the shape operator A A of the real hypersurface, for any nonnull real number k k . Following Tachibana [Analytic tensor and its generalization, Tohoku Math. J. 12 (1960), 208–221], we study the purity and hybridness of such tensors with respect to A A .

Pseudo‐Ricci–Yamabe solitons on real hypersurfaces in the complex quadric

AbstractFirst, we introduce a new notion of pseudo‐anti commuting for real hypersurfaces in the complex quadric and give a complete classification of Hopf pseudo‐Ricci–Yamabe soliton real hypersurfaces in the complex quadric . Next as an application we obtain a classification of gradient pseudo‐Ricci–Yamabe solitons on Hopf real hypersurfaces in .

Real hypersurfaces in nonflat complex space forms whose h-operator is parallel with respect to the generalized Tanaka-Webster connection

Real hypersurfaces in nonflat complex space forms whose h-operator is parallel with respect to the generalized Tanaka-Webster connection

Weakly $\phi$-invariant real hypersurfaces of dimension three

Weakly $\phi$-invariant real hypersurfaces of dimension three

Levi-flat real hypersurfaces in nonflat complex planes

Let M be a non-ruled weakly 2-Hopf real hypersurface in a nonflat complex plane whose mean curvature is invariant along the Reeb flow. In this paper, it is proved that M is Levi-flat if and only if M is locally congruent to a strongly 2-Hopf real hypersurface of a special type. As a corollary we present a class of non-ruled Levi-flat real hypersurfaces of dimension three with non-constant mean curvature.

On Real Analytic Levi-Flat Hypersurfaces Associated with Milnor Fibrations

On Real Analytic Levi-Flat Hypersurfaces Associated with Milnor Fibrations

Derivatives of structure Jacobi operator on real hypersurfaces in complex Grassmannians of rank two

Derivatives of structure Jacobi operator on real hypersurfaces in complex Grassmannians of rank two

The Shape Operator of Real Hypersurfaces in S6(1)

The Shape Operator of Real Hypersurfaces in S6(1)

New Results on Derivatives of the Shape Operator of Real Hypersurfaces in the Complex Quadric

A real hypersurface $M$ in the complex quadric $Q^{m}=SO_{m+2}/SO_mSO_2$ inherits an almost contact metric structure . This structure allows to define, for any nonnull real number $k$, the so called $k$-th generalized Tanaka-Webster connection on $M$, $\hat{\nabla}^{(k)}$. If $\nabla$ denotes the Levi-Civita connection on $M$, we introduce the concepts of $(\hat{\nabla}^{(k)},\nabla)$-Codazzi and $(\hat{\nabla}^{(k)},\nabla)$-Killing shape operator $S$ of the real hypersurface and classify real hypersurfaces in $Q$ satisfying any of these conditions.

Global Newlander–Nirenberg Theorem for Domains with C2 Boundary

The Newlander–Nirenberg theorem says that a formally integrable complex structure is locally equivalent to the standard complex structure in the complex Euclidean space. In this paper, we consider two natural generalizations of the Newlander–Nirenberg theorem under the presence of a C2 strictly pseudoconvex boundary. When a given formally integrable complex structure X is defined on the closure of a bounded strictly pseudoconvex domain with C2 boundary D⊂Cn, we show the existence of global holomorphic coordinate systems defined on D‾ that transform X into the standard complex structure provided that X is sufficiently close to the standard complex structure. Moreover, we show that such closeness is stable under a small C2 perturbation of ∂D. As a consequence, when a given formally integrable complex structure is defined on a one-sided neighborhood of some point in a C2 real hypersurface M⊂Cn, we prove the existence of local one-sided holomorphic coordinate systems provided that M is strictly pseudoconvex with respect to the given complex structure. We also obtain results when the structures are finite smooth.

Realization of arbitrary Lie algebras by automorphisms of $\mathrm{CR}$ manifolds and symmetries of differential equations

For any finite-dimensional real Lie algebra $\mathfrak{h}$, we construct a germ of a real analytic hypersurface in complex space such that its Lie algebra of infinitesimal holomorphic automorphisms is isomorphic to $\mathfrak{h}$. For any $\mathfrak{h}$, we also construct a system of partial differential equations whose Lie algebra of symmetries is isomorphic to the complexification of the algebra $\mathfrak{h}$.

Trajectories on Real Hypersurfaces of Type (A) and Those on a Complex Projective Space

Trajectories on Real Hypersurfaces of Type (A) and Those on a Complex Projective Space

Ruled real hypersurfaces in the complex hyperbolic quadric

Abstract In this article, we introduce a new family of real hypersurfaces in the complex hyperbolic quadric Q n ∗ = S O 2 , n o ∕ S O 2 S O n {{Q}^{n}}^{\ast }=S{O}_{2,n}^{o}/S{O}_{2}S{O}_{n} , namely, the ruled real hypersurfaces foliated by complex hypersurfaces. Berndt described an example of such a real hypersurface in Q n ∗ {{Q}^{n}}^{\ast } as a homogeneous real hypersurface generated by a A {\mathfrak{A}} -principal horocycle in a real form R H n {\mathbb{R}}{H}^{n} . So, in this article, we compute a detailed expression of the shape operator for ruled real hypersurfaces in Q n ∗ {{Q}^{n}}^{\ast } and investigate their characterizations in terms of the shape operator and the integrable distribution C = { X ∈ T M ∣ X ⊥ ξ } {\mathcal{C}}=\left\{X\in TM| X\perp \xi \right\} . Then, by using these observations, we give two kinds of classifications of real hypersurfaces in Q n ∗ {{Q}^{n}}^{\ast } satisfying η \eta -parallelism under either η \eta -commutativity of the shape operator or integrability of the distribution C {\mathcal{C}} . Moreover, we prove that the unit normal vector field of a real hypersurface with η \eta -parallel shape operator in Q n ∗ {{Q}^{n}}^{\ast } is A {\mathfrak{A}} -principal. On the other hand, it is known that all contact real hypersurfaces in Q n ∗ {{Q}^{n}}^{\ast } have a A {\mathfrak{A}} -principal normal vector field. Motivated by these results, we give a characterization of contact real hypersurfaces in Q n ∗ {{Q}^{n}}^{\ast } in terms of η \eta -parallel shape operator.

On the Structure Lie Operator of a Real Hypersurface in the Complex Quadric

ABSTRACT The almost contact metric structure that we have on a real hypersurface M in the complex quadric Qm = SO m+2/SO m SO 2 allows us to define, for any nonnull real number k, the k-th generalized Tanaka-Webster connection on M, ∇ ^ ( k ) . Associated to this connection, we have Cho and torsion operators F X ( k ) and T X ( k ) , respectively, for any vector field X tangent to M. From them and for any symmetric operator B on M, we can consider two tensor fields of type (1,2) on M that we denote by B F ( k ) and B T ( k ) , respectively. We classify real hypersurfaces M in Qm for which any of those tensors identically vanishes, in the particular case of B being the structure Lie operator Lξ on M.

Cyclic Parallel Structure Jacobi Operator for Real Hypersurfaces in the Complex Quadric

Cyclic Parallel Structure Jacobi Operator for Real Hypersurfaces in the Complex Quadric

The Probabilistic Method in Real Singularity Theory

We explain how to use the probabilistic method to prove the existence of real polynomial singularities with rich topology, i.e., with total Betti number of the maximal possible order. We show how similar ideas can be used to produce real algebraic projective hypersurfaces with a rich structure of umbilical points.

Parallelism of Structure Lie Operators on Real Hypersurfaces in Nonflat Complex Space Forms

Parallelism of Structure Lie Operators on Real Hypersurfaces in Nonflat Complex Space Forms