arXiv research

Study on submanifolds in metallic structures with new results and structures.

The study explores slant submanifolds in Golden Riemannian manifolds.

Study CR-submanifolds in generalized Kähler manifolds.

Study metallic Riemannian structures on submanifolds of Riemannian manifolds.

Study of CR-submanifolds in various Lorentzian manifolds.

The study characterizes submanifolds in product spaces.

The paper studies geometric representations of submanifolds using complex-valued functions.

We write down the local equations that characterize the submanifolds N of a Dirac manifold M which have a normal bundle that is either a coisotropic or an isotropic submanifold of TM endowed with the tangent Dirac structure. In the Poisson case, these formulas prove again a result of Xu: the submanifold N has a normal …

Let X be a compact Kahler manifold with a non-trivial holomorphic Poisson structure. Then there exist deformations of non-trivial generalized Kahler structures with one pure spinor on X. We prove that every Poisson submanifold of X is a generalized Kahler submanifold with respect to the deformed generalized Kahler stru…

We give some fundamental properties of the induced structures on submanifolds immersed in almost product or locally product Riemannian manifolds. We study the induced structure by the composition of two isometric immersions on submanifolds in an almost product Riemannian manifold. We give an effective construction for …

Study on slant submanifolds with new conditions and transitivity.

Characterizes blowups of Dirac structures on manifolds.

The paper establishes a connection between superminimal surfaces and Lagrangian submanifolds in twistor spaces.

Estimates tangent space variation on Riemannian submanifolds using feature size.

Submanifolds of Frobenius manifolds are studied. In particular, so-called natural submanifolds are defined and, for semi-simple Frobenius manifolds, classified. These carry the structure of a Frobenius algebra on each tangent space, but will, in general, be curved. The induced curvature is studied, a main result being …

Study neighbourhoods of submanifolds in generalized complex geometry.

The paper explores deformations of compact holomorphic Poisson submanifolds.

In generalized complex geometry, we revisit linear subspaces and submanifolds that have an induced generalized complex structure. We give an expression of the induced structure that allows us to deduce a smoothness criteria, we dualize the results to submersions and we make a few comments on generalized complex mapping…

Study minimal Lagrangian submanifolds of complex hyperquadric.

Study CR submanifolds of nearly Kähler S³ × S³ using properties of its almost product structure.

An n-dimensional submanifold X of a projective space P^N (C) is called tangentially degenerate if the rank of its Gauss mapping γ: X ---> G (n, N) satisfies 0 < rank γ< n. The authors systematically study the geometry of tangentially degenerate submanifolds of a projective space $P^N (\mathbf{C})$. By means of the foca…

We introduce the notion of twisted generalized complex submanifolds and describe an equivalent characterization in terms of Poisson-Dirac submanifolds. Our characterization recovers a result of Vaisman. An equivalent characterization is also given in terms of spinors. As a consequence, we show that the fixed locus of a…

We introduce a class of potential submanifolds in pseudo-Euclidean spaces (each N-dimensional potential submanifold is a special flat torsionless submanifold in a 2N-dimensional pseudo-Euclidean space) and prove that each N-dimensional Frobenius manifold can be locally represented as an N-dimensional potential submanif…

Study of mean curvature flow in hyperkähler manifolds leading to complex Lagrangian submanifolds.

Study characterizes submanifolds in metallic semi-Riemannian manifolds with specific connections.

New formulas for mean curvature of submanifolds in geometries with torsion.

We discuss various algebraic quantum structures associated to monotone Lagrangian submanifolds and we present a number of applications, computations and examples.

We show that there exist infinitely many pairwise distinct non-closed G_2-manifolds (some of which have holonomy full G_2) such that they admit co-oriented contact structures and have co-oriented contact submanifolds which are also associative. Along the way, we prove that there exists a tubular neighborhood N of every…

We prove that the associativity equations of two-dimensional topological quantum field theories are very natural reductions of the fundamental nonlinear equations of the theory of submanifolds in pseudo-Euclidean spaces and give a natural class of potential flat torsionless submanifolds. We show that all potential flat…

Let M be an 8-manifold with a Spin(7)-structure. We first show that closed Cayley submanifolds of M form a smooth moduli space for a generic Spin(7)-structure. Then we study the deformations of a compact, connected Cayley submanifold X of M with non-empty boundary contained in a given submanifold W of M such that X and…

The paper studies deformations of submanifolds using a new algebraic structure.

A conformal structure on a manifold $M^n$ induces natural second order conformally invariant operators, called Möbius and Laplace structures, acting on specific weight bundles of $M$, provided that $n\ge 3$. By extending the notions of Möbius and Laplace structures to the case of surfaces and curves, we develop here th…

The paper describes Calabi-Yau structures and special Lagrangian submanifolds in complexified symmetric spaces.

Diffeological submanifolds are a new type of submanifold in manifold theory.

The following results are proved: Theorem 1. A totally real semiparallel submanifold of constant curvature with parallel f-structure in the normal bundle of a Kähler manifold N is flat or a totally geodesic submanifold of N. Theorem 2. A totally real minimal semiparallel submanifold M with parallel f-structure in the n…

Explores local structure of morphisms and formal submanifolds in formal manifolds theory.

We give explicit examples of degree 3 cohomology classes not Poincare dual to submanifolds, and discuss the realisability of homology classes by submanifolds with Spin-C normal bundles.

The study explores properties of slant and semi-slant submanifolds in metallic Riemannian manifolds.

Invariant submanifolds maintain constant angles with Reeb vector fields in metric contact pairs.

3D projective structures can be metrized with conformal structures.

Constructs Lorentzian manifolds from Riemannian conformal structures.

The study describes the structure at infinity of submanifolds in real space forms.

This study addresses transitions in conically singular associative submanifolds and their desingularizations.

We study a natural Dirac operator on a Lagrangian submanifold of a Kähler manifold. We first show that its square coincides with the Hodge-de Rham Laplacian provided the complex structure identifies the Spin structures of the tangent and normal bundles of the submanifold. We then give extrinsic estimates for the eigenv…

Study on special null submanifolds in indefinite Sasakian manifolds.

Study submanifolds in Koszul-Vinberg geometry, a blend of Poisson and pseudo-Riemannian structures.

We study isoparametric submanifolds of rank at least two in a separable Hilbert space, which are known to be homogeneous by a result of Heintze and Liu, and associate to such a submanifold M and a point x in M a canonical homogeneous structure (a certain bilinear map on the tangent space). We prove that the homogeneous…

The notion of a Frobenius submanifold - a submanifold of a Frobenius manifold which is itself a Frobenius manifold with respect to structures induced from the original manifold - is studied. Two dimensional submanifolds are particularly simple. More generally, sufficient conditions are given for a submanifold to be a s…