Class Of Submanifolds Research Articles

STCR-Lightlike Product Manifolds of an Indefinite Kaehler Statistical Manifold with a Quarter Symmetric Non-Metric Connection

The present work aims to introduce a novel class of submanifolds, namely STCR-lightlike submanifolds, for an indefinite Kaehler statistical manifold with a quarter symmetric non-metric connection. The characterization theorems on totally umbilical and totally geodesic STCR-lightlike submanifolds with respect to the integrability of distributions have been established. Some conditions for a STCR-lightlike submanifold to be a STCR-lightlike product manifold have been derived.

Conformal Kaehler Euclidean submanifolds

Let f:M2n→R2n+ℓ, n≥5, be a conformal immersion into Euclidean space with codimension ℓ where M2n is a Kaehler manifold of complex dimension n free of points where all sectional curvatures vanish. For codimension ℓ=1 or ℓ=2 we show that at least locally such a submanifold can always be obtained in a rather simple way, namely, from an isometric immersion of the Kaehler manifold M2n into either R2n+1 or R2n+2, the latter being a class of submanifolds already extensively studied.

Kac-Rice formula for transverse intersections

We prove a generalized Kac-Rice formula that, in a well defined regular setting, computes the expected cardinality of the preimage of a submanifold via a random map by expressing it as the integral of a density. Our proof starts from scratch and although it follows the guidelines of the standard proofs of Kac-Rice formula, it contains some new ideas coming from the point of view of measure theory. Generalizing further, we extend this formula to any other type of counting measure, such as the intersection degree. We discuss in depth the specialization to smooth Gaussian random sections of a vector bundle. Here, the formula computes the expected number of points where the section meets a given submanifold of the total space, it holds under natural non-degeneracy conditions and can be simplified by using appropriate connections. Moreover, we point out a class of submanifolds, that we call sub-Gaussian, for which the formula is locally finite and depends continuously with respect to the covariance of the first jet. In particular, this applies to any notion of singularity of sections that can be defined as the set of points where the jet prolongation meets a given semialgebraic submanifold of the jet space. Various examples of applications and special cases are discussed. In particular, we report a new proof of the Poincaré kinematic formula for homogeneous spaces and we observe how the formula simplifies for isotropic Gaussian fields on the sphere.

On pointwise quasi bi-slant submanifolds

In this paper, we introduce a new class of submanifolds which are called pointwise quasi bi-slant submanifolds in almost Hermitian manifolds which extends quasi bi-slant, bi-slant, hemi-slant, semi-slant and slant submanifolds in a very natural way. Several basic results in this respect are proved in this paper. Moreover, we obtain some conditions of the distributions which is involved in the definition of the new submanifolds. We also get some results for totally geodesic and mixed totally geodesic conditions for pointwise quasi bi-slant submanifolds. Finally, we illustrate some examples in order to guarantee the new kind of submanifolds.

Chen’s [formula omitted]-invariants type inequalities for bi-slant submanifolds in generalized Sasakian space forms

In this paper, we establish optimal inequalities involving generalized δ-Casorati curvature δC(k;s−1) for the bi-slant submanifolds of generalized Sasakian space forms endowed with a quarter-symmetric connection. The equality case is discussed for ideal submanifolds. Furthermore, the special cases of derived inequality is given for C-totally real submanifolds and some other classes of submanifolds.

Sparse Data Interpolation and Smoothing on Embedded Submanifolds

Energy minimization is one of the properties that make univariate splines so favorable in many problems of approximation and estimation; interpolation in and extrapolation from sparse data sites and smoothing of noisy data in particular. In this paper, we present a novel approach to approximate energy minimization on certain classes of submanifolds that gives rise to new methods for extrapolation and smoothing on submanifolds. To accomplish this, we minimize intrinsic functionals approximately by minimising a suitable extrinsic formulation of the functional augmented by a penalty on the first order normal derivative. The general framework we develop is accompanied by error analysis and exemplified by tensor product B-splines.

Self-shrinker Type Submanifolds in the Euclidean Space

In this paper, we devote to investigating a new class of submanifolds, named the self-shrinker type submanifolds. We define a functional and prove its critical point is a self-shrinker type submanifold. We also show that a compact self-shrinker type submanifold with the parallel mean curvature vector in the Euclidean Space $$\mathbb {R}^{n+p}$$ is a minimal submanifold in a hypersphere $$\mathbb {S}^{n+p-1}$$ of $$\mathbb {R}^{n+p}$$ .

On a Class of Submanifolds in a Tangent Bundle with a g-Natural Metric - Normal lift

On a Class of Submanifolds in a Tangent Bundle with a g-Natural Metric - Normal lift

An optimal inequality for Lagrangian submanifolds in complex space forms involving Casorati curvature

In this paper, we establish an optimal inequality involving normalized δ-Casorati curvature δC(n−1) of Lagrangian submanifolds in n-dimensional complex space forms. We derive a very singular and unexpected result: the lower bounds of the normalized δ-Casorati curvatures δC(n−1) and δCˆ(n−1) in terms of dimension, the holomorphic sectional curvature, the normalized scalar curvature and the squared mean curvature of the submanifold, are different, in contrast to all previous results obtained for several classes of submanifolds in many ambient spaces. We also investigate the equality case of the inequality and prove that a Casorati δC(n−1)-ideal Lagrangian submanifold of a complex space form without totally geodesic points is an H-umbilical Lagrangian submanifold of ratio 4. Some examples are discussed in the last part of the paper, showing that the constants in the inequality obtained in this work are the best possible.

The Vectorial Ribaucour Transformation for Submanifolds of Constant Sectional Curvature

We obtain a reduction of the vectorial Ribaucour transformation that preserves the class of submanifolds of constant sectional curvature of space forms, which we call the L-transformation. It allows to construct a family of such submanifolds starting with a given one and a vector-valued solution of a system of linear partial differential equations. We prove a decomposition theorem for the L-transformation, which is a far-reaching generalization of the classical permutability formula for the Ribaucour transformation of surfaces of constant curvature in Euclidean three space. As a consequence, we derive a Bianchi-cube theorem, which allows to produce, from k initial scalar L-transforms of a given submanifold of constant curvature, a whole k-dimensional cube all of whose remaining $$2^k-(k+1)$$ vertices are submanifolds with the same constant sectional curvature given by explicit algebraic formulae. We also obtain further reductions, as well as corresponding decomposition and Bianchi-cube theorems, for the classes of n-dimensional flat Lagrangian submanifolds of $${\mathbb {C}}^n$$ and n-dimensional Lagrangian submanifolds with constant curvature c of the complex projective space $${\mathbb {C}}{\mathbb {P}}^n(4c)$$ or the complex hyperbolic space $${\mathbb {C}}{\mathbb {H}}^n(4c)$$ of complex dimension n and constant holomorphic curvature 4c.

On a class of submanifolds in a tangent bundle with a $g$-natural metric

On a class of submanifolds in a tangent bundle with a $g$-natural metric

Certain condition on the second fundamental form of CR submanifolds of maximal CR dimension of complex hyperbolic space

We treat m-dimensional real submanifolds M of complex space forms M when the maximal holomorphic tangent subspace is (m−1)-dimensional. On these manifolds there exists an almost contact structure F which is naturally induced from the ambient space and in this paper we study the condition h(FX,Y)−h(X,FY) = g(FX,Y)η, η∊ T⊥(M), on the structure F and on the second fundamental form h of these submanifolds. Especially when the ambient space M is a complex Euclidean space, we obtain a complete classification of submanifolds M which satisfy these conditions.

HAMILTONIAN FIELDS AND ENERGY IN CONTACT MANIFOLDS

To the contact distribution of a contact manifold we associate Hamiltonian type vector fields, called contact Hamiltonian fields. Their properties are investigated and the existence of such vector fields nowhere tangent to a given submanifold is proved. Time-depending contact Hamiltonian vector fields allow us to define the contact energy whose properties are studied. A class of submanifolds in relation to the study of contact Hamiltonian fields is also analyzed.

Tableaux over Lie algebras, integrable systems, and classical surface theory

Starting from suitable tableaux over finite dimensional Lie algebras, we provide a scheme for producing involutive linear Pfaffian systems related to various classes of submanifolds in homogeneous spaces which constitute integrable systems. These include isothermic surfaces, Willmore surfaces, and other classical soliton surfaces. Completely integrable equations such as the G/G(o)-system of Terng and the curved flat system of Ferus-Pedit may be obtained as special cases of this construction. Some classes of surfaces in projective differential geometry whose Gauss-Codazzi equations are associated with tableaux over sl(4, R) are discussed

TANGENT DIRAC STRUCTURES AND SUBMANIFOLDS

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 once again prove a result of Xu: the submanifold N has a normal bundle which is a coisotropic submanifold of TM with the tangent Poisson structure if and only if N is a Dirac submanifold. In the presymplectic case it is the isotropy of the normal bundle which characterizes the corresponding notion of a Dirac submanifold. On the way, we give a simple definition of the tangent Dirac structure, make new remarks about it and establish its characteristic, local formulas for various interesting classes of submanifolds of a Dirac manifold.

On a class of submanifolds carrying an extrinsic totally umbilical foliation

In this paper we give a conformal classification of all euclidean submanifolds carrying a parallel principal curvature normal under the intrinsic additional assumption that the associated conformal conullity is involutive and the leaves are extrinsic spheres in the submanifold in the sense of Nomizu. We also provide several applications of this result.

Integrability of implicit differential equations

The problem of integrability of differential equations is discussed. Examples and integrability criteria are given. An algorithm for extracting the integrable part of an implicit differential equation is formulated. A procedure for generating a class of submanifolds of the cotangent bundle is defined. This procedure is then used for generating implicit differential equations in the phase space of a mechanical system. Integrability criteria of such equations are established and an extraction algorithm is formulated.

Ric-semisymmetric submanifolds

This article surveys results obtained on various classes of submanifolds with the interior geometry of Riccisemisymmetric spaces.

Infinitesimally homogeneous submanifolds of euclidean spaces

For a class of submanifolds of ℝN, the infinitesimally homogeneous ones, the second fundamental form and itss-times iterated derivativess≤k+1 at any fixed point determine the immersion uniquely. The integerk>0 will be called the extrinsic Singer invariant. Any infinitesimally homogeneous submanifoldM (which is not necessarily complete) is an open part of a globally homogeneous (complete) submanifold. Indeed, the infinitesimal data at any pointp, determine, canonically, a Lie subgroupG of the isometry group of ℝ N , whose orbit atp is a complete submanifold that extendsM.

Holonomy group and generalized Maslov classes of submanifolds of spaces with an affine connection

Holonomy group and generalized Maslov classes of submanifolds of spaces with an affine connection