Real Submanifolds Research Articles

Chern–Moser operators and weighted jet determination problems in higher codimension

Counterexamples to the [Formula: see text]-jet determination Chern–Moser Theorem in codimension [Formula: see text] have recently been constructed [16]. We extend the Chern–Moser approach for hypersurfaces to real submanifolds of higher codimension in complex space to derive results on jet determination for their automorphism group. Using these techniques, we show that the [Formula: see text]-jet determination Chern–Moser Theorem holds in codimension 2.

Pullback Coherent States, Squeezed States and Quantization

In this semi-expository paper, we define certain Rawnsley-type coherent and squeezed states on an integral K\"ahler manifold (after possibly removing a set of measure zero) and show that they satisfy some properties which are akin to maximal likelihood property, reproducing kernel property, generalised resolution of identity property and overcompleteness. This is a generalization of a result by Spera. Next we define the Rawnsley-type pullback coherent and squeezed states on a smooth compact manifold (after possibly removing a set of measure zero) and show that they satisfy similar properties. Finally we show a Berezin-type quantization involving certain operators acting on a Hilbert space on a compact smooth totally real embedded submanifold of $U$ of real dimension $n$, where $U$ is an open set in ${\mathbb C}{\rm P}^n$. Any other submanifold for which the criterion of the identity theorem holds exhibit this type of Berezin quantization. Also this type of quantization holds for totally real submanifolds of real dimension $n$ of a general homogeneous K\"ahler manifold of real dimension $2n$ for which Berezin quantization exists. In the appendix we review the Rawnsley and generalized Perelomov coherent states on ${\mathbb C}{\rm P}^n$ (which is a coadjoint orbit) and the fact that these two types of coherent states coincide.

The Baouendi-Treves approximation theorem for Gevrey classes and applications

In this work we show how to extend the seminal Baouendi-Treves approximation theorem for Gevrey functions and ultradistributions. As applications we present a Gevrey version of the approximate Poincar\'e Lemma and study ultradistributions vanishing on maximally real submanifolds.

The finite jet determination problem for CR maps of positive codimension into Nash manifolds

We prove the first general finite jet determination result in positive codimension for CR maps from real-analytic minimal submanifolds M ⊂ C N $M\subset \mathbb {C}^N$ into Nash (real) submanifolds M ′ ⊂ C N ′ $M^{\prime }\subset \mathbb {C}^{N^{\prime }}$ . For a sheaf S $\mathcal {S}$ of C ∞ $\mathcal {C}^\infty$ -smooth CR maps from M $M$ into M ′ $M^{\prime }$ , we show that the non-existence of so-called 2-approximate CR S $\mathcal {S}$ -deformations from M $M$ into M ′ $M^{\prime }$ implies the following strong finite jet determination property: There exists a map ℓ : M → Z + $\ell \colon M\rightarrow {\mathbb {Z}}_+$ , bounded on compact subsets of M $M$ , such that for every point p ∈ M $p\in M$ , whenever f , g $f,g$ are two elements of S p $\mathcal {S}_p$ with j p ℓ ( p ) f = j p ℓ ( p ) g $j^{\ell (p)}_pf=j^{\ell (p)}_pg$ , then f = g $f=g$ . Applying the deformation point of view allows a unified treatment of a number of classes of target manifolds, which includes, among others, strictly pseudoconvex, Levi–non-degenerate, but also some particularly important Levi-degenerate targets, such as boundaries of classical domains.

Construction of Networks by Associating with Submanifolds of Almost Hermitian Manifolds

The idea that data lies in a non-linear space has brought up the concept of manifold learning as a part of machine learning and such notion is one of the most important research fields of today. The main idea here is to design the data as a submanifold model embedded in a high-dimensional manifold. On the other hand, graph theory is one of the most important research areas of applied mathematics and computer science. As a result, many researchers investigate new methods for machine learning on graphs. From the above information, it is seen that the theory of submanifolds and graph theory have become two important concepts in machine learning and nowadays, the geometric deep learning research area using these two concepts has emerged. By combining these two fields, this article aims to present the relationships between submanifolds of complex manifolds with the help of graphs. In this paper, we build some directed networks by identifying with submanifolds of almost Hermitian manifolds. Moreover, we give some results and relations among holomorphic submanifolds, totally real submanifolds, CR-submanifolds, slant submanifolds, semi-slant submanifolds, hemi-slant submanifolds, and bi-slant submanifolds in almost Hermitian manifolds in terms of graph theory.

Chen Inequalities for Spacelike Submanifolds in Statistical Manifolds of Type Para-Kähler Space Forms

In this paper, we prove some inequalities between intrinsic and extrinsic curvature invariants, namely involving the Chen first invariant and the mean curvature of totally real and holomorphic spacelike submanifolds in statistical manifolds of type para-Kähler space forms. Furthermore, we investigate the equality cases of these inequalities. As illustrations of the applications of the above inequalities, we consider a few examples.

Minimal real Kaehler submanifolds

We show that generic rank conditions on the second fundamental form of an isometric immersion $f\colon M^{2n}\to\mathbb{R}^{2n+p}$ of a Kaehler manifold of complex dimension $n\geq 2$ into Euclidean space with low codimension $p$ implies that the submanifold has to be minimal. If $M^{2n}$ if simply connected, this amounts to the existence of a one-parameter associated family of isometric minimal immersions unless $f$ is holomorphic.

A characterization for totally real submanifolds using self-adjoint differential operator

<abstract><p>In this article, we study totally real submanifolds in Kaehler product manifold with constant scalar curvature using self-adjoint differential operator $ \Box $. Under this setup, we obtain a characterization result. Moreover, we discuss $ \delta- $invariant properties of such submanifolds and get an obstruction result as an application of the inequality derived. The results in the article are supported by non-trivial examples.</p></abstract>

A characterization of totally real statistical submanifolds in quaternion Kaehler-like statistical manifolds

A characterization of totally real statistical submanifolds in quaternion Kaehler-like statistical manifolds

The eigenvalue estimates of p-Laplacian of totally real submanifolds in generalized complex space forms

The eigenvalue estimates of p-Laplacian of totally real submanifolds in generalized complex space forms

Inequalities for the Casorati Curvature of Totally Real Spacelike Submanifolds in Statistical Manifolds of Type Para-Kähler Space Forms.

The purpose of this article is to establish some inequalities concerning the normalized -Casorati curvatures (extrinsic invariants) and the scalar curvature (intrinsic invariant) of totally real spacelike submanifolds in statistical manifolds of the type para-Kähler space form. Moreover, this study is focused on the equality cases in these inequalities. Some examples are also provided.

A Splitting Result for Real Submanifolds of a Kähler Manifold

Let (Z,omega ) be a connected Kähler manifold with an holomorphic action of the complex reductive Lie group U^mathbb {C}, where U is a compact connected Lie group acting in a hamiltonian fashion. Let G be a closed compatible Lie group of U^mathbb {C} and let M be a G-invariant connected submanifold of Z. Let xin M. If G is a real form of U^mathbb {C}, we investigate conditions such that Gcdot x compact implies U^mathbb {C}cdot x is compact as well. The vice-versa is also investigated. We also characterize G-invariant real submanifolds such that the norm-square of the gradient map is constant. As an application, we prove a splitting result for real connected submanifolds of (Z,omega ) generalizing a result proved in Gori and Podestà (Ann Global Anal Geom 26: 315–318, 2004), see also Bedulli and Gori (Results Math 47: 194–198, 2005), Biliotti (Bull Belg Math Soc Simon Stevin 16: 107–116 2009).

Remarks on conformal anti-invariant Riemannian maps to cosymplectic manifolds

M.A. Akyol and B. Şahin [Conformal anti-invariant Riemannian maps to Kaehler manifolds, U.P.B. Sci. Bull., Series A, Vol. 80, Iss. 4, 2018] defined and studied the notion of conformal anti-invariant Riemannian maps to Kaehler manifolds. In this paper, as a generalization of totally real submanifolds and anti-invariant Riemannian maps, we extend this notion to almost contact metric manifolds. In this manner, we introduce conformal anti-invariant Riemannian maps from Riemannian manifolds to cosymplectic manifolds. In order to guarantee the existence of this notion, we give a non-trivial example, investigate the geometry of foliations which are arisen from the definition of a conformal Riemannian map and obtain decomposition theorems by using the existence of conformal Riemannian maps. Moreover, we investigate the harmonicity of such maps and find necessary and sufficient conditions for conformal anti-invariant Riemannian maps to be totally geodesic. Finally, we study weakly umbilical conformal Riemannian maps and obtain a classification theorem for conformal anti-invariant Riemannian maps.

C-Totally Real Submanifolds with Constant Sectional Curvature in the Sasakian Space Forms

C-Totally Real Submanifolds with Constant Sectional Curvature in the Sasakian Space Forms

Instability of rational and polynomial convexity

It is shown that rational and polynomial convexity of totally real submanifolds is in general unstable under perturbations that are C α C^\alpha -small for any Hölder exponent α > 1 \alpha >1 . This complements the result of Løw and Wold that these properties are C 1 C^1 -stable.

An investigation on the existence of warped product irrotational screen-real lightlike submanifolds of metallic semi-Riemannian manifolds

PurposeThe purpose of this paper is to study the geometry of screen real lightlike submanifolds of metallic semi-Riemannian manifolds. Also, the authors investigate whether these submanifolds are warped product lightlike submanifolds or not.Design/methodology/approachThe paper is design as follows: In Section 3, the authors introduce screen-real lightlike submanifold of metallic semi Riemannian manifold. In Section 4, the sufficient conditions for the radical and screen distribution of screen-real lightlike submanifolds, to be integrable and to be have totally geodesic foliation, have been established. Furthermore, the authors investigate whether these submanifolds can be written in the form of warped product lightlike submanifolds or not.FindingsThe geometry of the screen-real lightlike submanifolds has been studied. Also various results have been established. It has been proved that there does not exist any class of irrotational screen-real r-lightlike submanifold such that it can be written in the form of warped product lightlike submanifolds.Originality/valueAll results are novel and contribute to further study on lightlike submanifolds of metallic semi-Riemannian manifolds.

On equivalence problem for 2–nondegenerate CR geometries with simple models

In this article, we solve the equivalence problem for 2–nondegenerate CR geometries that have (at every point) a homogeneous space G/H as a maximally symmetric model for G simple real Lie group of CR automorphisms. This completes the classification of real submanifolds in complex space that are maximally symmetric models with a real simple CR automorphism group. In particular, we construct (local) embeddings of these models into complex space.

Addendum to “Stationary discs and finite jet determination for non-degenerate generic real submanifolds” [Adv. Math. 343 (2019) 910–934

We previously introduced a new notion of non-degeneracy for generic real submanifolds in CN. The definition is however not complete and the purpose of this addendum is to complete it.

The Chekanov torus in $S^2 \times S^2$ is not real

We prove that the count of Maslov index 2 $J$-holomorphic discs passing through a generic point of a real Lagrangian submanifold in a closed spherically monotone symplectic manifold must be even. As a corollary, we exhibit a genuine real symplectic phenomenon in terms of involutions, namely that the Chekanov torus $\mathbb{T}_{\text{Chek}}$ in $S^2\times S^2$, which is a monotone Lagrangian torus not Hamiltonian isotopic to the Clifford torus $\mathbb{T}_{\text{Clif}}$, can be seen as the fixed point set of a smooth involution, but not of an antisymplectic involution.

Deformations of CR maps and applications

We study the deformation theory of CR maps in the positive codimensional case. In particular, we study structural properties of the {\em mapping locus} $E$ of (germs of nondegenerate) holomorphic maps $H \colon (M,p) \to M'$ between generic real submanifolds $M \subset \mathbb C^N$ and $M' \subset \mathbb C^{N'}$, defined to be the set of points $p' \in M'$ which admit such a map with $H(p) = p'$. We show that this set $E$ is semi-analytic and provide examples for which $E$ posseses (prescribed) singularities.