Geodesic Submanifolds Research Articles

Polynomial chaos expansions on principal geodesic Grassmannian submanifolds for surrogate modeling and uncertainty quantification

In this work we introduce a manifold learning-based surrogate modeling framework for uncertainty quantification in high-dimensional stochastic systems. Our first goal is to perform data mining on the available simulation data to identify a set of low-dimensional (latent) descriptors that efficiently parameterize the response of the high-dimensional computational model. To this end, we employ Principal Geodesic Analysis on the Grassmann manifold of the response to identify a set of disjoint principal geodesic submanifolds, of possibly different dimension, that captures the variation in the data. Since operations on the Grassmann require the data to be concentrated, we propose an adaptive algorithm based on Riemannian K-means and the minimization of the sample Fréchet variance on the Grassmann manifold to identify “local” principal geodesic submanifolds that represent different system behavior across the parameter space. Polynomial chaos expansion is then used to construct a mapping between the random input parameters and the projection of the response on these local principal geodesic submanifolds. The method is demonstrated on four test cases, a toy-example that involves points on a hypersphere, a Lotka-Volterra dynamical system, a continuous-flow stirred-tank chemical reactor system, and a two-dimensional Rayleigh-Bénard convection problem.

On the injectivity of the shifted Funk–Radon transform and related harmonic analysis

AbstractNecessary and sufficient conditions are obtained for injectivity of the shifted Funk–Radon transform associated with k-dimensional totally geodesic submanifolds of the unit sphere Sn in ℝn+1. This result generalizes the well known statement for the spherical means on Sn and is formulated in terms of zeros of Jacobi polynomials. The relevant harmonic analysis is developed, including a new concept of induced Stiefel (or Grassmannian) harmonics, the Funk–Hecke type theorems, addition formula, and multipliers. Some perspectives and conjectures are discussed.

Nondivergence on homogeneous spaces and rigid totally geodesic submanifolds

Nondivergence on homogeneous spaces and rigid totally geodesic submanifolds

Multiplicative rectifying submanifolds of multiplicative Euclidean space

In this paper, we initiate the study of the multiplicative Euclidean submanifolds, exhibiting the multiplicative counterparts of the Gauss–Weingarten formulas. Some particular types such as multiplicative conic, spherical, and totally geodesic submanifolds are analyzed. In relation to such multiplicative submanifolds, we introduce multiplicative rectifying submanifolds and characterize by means of multiplicative concurrent vector fields. The multiplicative rectifying hypersurfaces are also obtained.

Totally geodesic submanifolds in products of rank one symmetric spaces

Totally geodesic submanifolds in products of rank one symmetric spaces

Chen–Ricci Inequality for Isotropic Submanifolds in Locally Metallic Product Space Forms

In this article, we study isotropic submanifolds in locally metallic product space forms. Firstly, we establish the Chen–Ricci inequality for such submanifolds and determine the conditions under which the inequality becomes equality. Additionally, we explore the minimality of Lagrangian submanifolds in locally metallic product space forms, and we apply the result to create a classification theorem for isotropic submanifolds whose mean curvature is constant. More specifically, we have demonstrated that the submanifolds are either a product of two Einstein manifolds with Einstein constants, or they are isometric to a totally geodesic submanifold. To support our findings, we provide several examples.

Totally geodesic Lagrangian submanifolds of the pseudo‐nearly Kähler SL(2,R)×SL(2,R)$\mathrm{SL}(2,\mathbb {R})\times \mathrm{SL}(2,\mathbb {R})$

AbstractIn this paper, we study Lagrangian submanifolds of the pseudo‐nearly Kähler . First, we show that they split into four different classes depending on their behavior with respect to a certain almost product structure on the ambient space. Then, we give a complete classification of totally geodesic Lagrangian submanifolds of this space.

On CR-lightlike submanifolds in a golden semi-Riemannian manifold

<abstract><p>CR-lightlike submanifolds of a golden semi-Riemannian manifold are the focus of the research presented in this work, which aims to define and investigate these structures. Under the context of a golden semi-Riemannian manifold, we study the properties of geodesic CR-lightlike submanifolds as well as umbilical CR-lightlike submanifolds. In addition, on a golden semi-Riemannian manifold, we find numerous intriguing findings for entirely geodesic and totally umbilical CR-lightlike subamnifolds. Also, we construct an example of a CR-lightlike submanifold of a golden semi-Riemannian manifold.</p></abstract>

Characterizations for pseudoparalel submanifolds of Lorentz-Sasakian space forms

In this article, total geodesic submanifolds for Lorentz-Sasakian space forms are investigated. For these submanifolds, pseudoparallel, 2-pseudoparallel, Ricci generalized pseudoparallel and 2-Ricci generalized pseudoparallel invariant submanifolds have been studied and many new results have been obtained. In addition, necessary and sufficient conditions have been obtained for these submanifolds to be total geodesic on the concircular and projective curvature tensors.

Ricci Soliton Lightlike Submanifolds with co-dimenison 2

The necessary requirements for half-lightlike and coisotropic lightlike submanifolds to be a Ricci soliton are obtained. Some examples of Ricci soliton half-lightlike and Ricci soliton coisotropic lightlike submanifolds are given. The Ricci soliton equation is investigated on totally geodesic, totally umbilical and irrotational lightlike submanifolds.

On gap rigidity problems for compact Hermitian symmetric spaces

We prove a gap rigidity theorem for diagonal curves in maximal polyspheres of irreducible compact Hermitian symmetric spaces of tube type, which is a dual analog to a theorem obtained by Mok. Motivated by the proof we show that the Chow space of certain totally geodesic submanifolds is affine algebraic, which gives a weaker version of gap rigidity for a class of higher dimensional submanifolds.

Totally geodesic submanifolds in exceptional symmetric spaces

We classify maximal totally geodesic submanifolds in exceptional symmetric spaces up to isometry. Moreover, we introduce an invariant for certain totally geodesic embeddings of semisimple symmetric spaces, which we call the Dynkin index. We prove a result analogous to the index conjecture: for every irreducible symmetric space of non-compact type, there exists a totally geodesic submanifold which is of minimal codimension and whose non-flat irreducible factors have Dynkin index equal to one.

Arithmeticity, superrigidity and totally geodesic submanifolds of complex hyperbolic manifolds

For $$n \ge 2$$ , we prove that a finite volume complex hyperbolic n-manifold containing infinitely many maximal properly immersed totally geodesic submanifolds of real dimension at least two is arithmetic, paralleling our previous work for real hyperbolic manifolds. As in the real hyperbolic case, our primary result is a superrigidity theorem for certain representations of complex hyperbolic lattices. The proof requires developing new general tools not needed in the real hyperbolic case. Our main results also have a number of other applications. For example, we prove nonexistence of certain maps between complex hyperbolic manifolds, which is related to a question of Siu, that certain hyperbolic 3-manifolds cannot be totally geodesic submanifolds of complex hyperbolic manifolds, and that arithmeticity of complex hyperbolic manifolds is detected purely by the topology of the underlying complex variety, which is related to a question of Margulis. Our results also provide some evidence for a conjecture of Klingler that is a broad generalization of the Zilber–Pink conjecture.

Real hypersurfaces foliated by totally real totally geodesic submanifolds

We bridge between submanifold geometry and curve theory. In the first half of this paper we classify real hypersurfaces in a complex projective plane and a complex hyperbolic plane all of whose integral curves γ of the characteristic vector field are totally real circles of the same curvature which is independent of the choice of γ in these planes. In the latter half, we construct real hypersurfaces which are foliated by totally real (Lagrangian) totally geodesic submanifolds in a complex hyperbolic plane, which provide one of the examples obtained in the classification.

Aq-components of geometric classes in compact Hermitian locally symmetric spaces

Let Γ∖G∕K be a compact Hermitian locally symmetric space, where G is simple. We study the components of a de Rham cohomology class of Γ∖G∕K with respect to the Matsushima decomposition, where the class is obtained by taking the Poincaré dual of a totally geodesic complex analytic submanifold. Using an improved version of the vanishing result of Kobayashi and Oda, we specify the existence of certain components of such cohomology classes when G=SU(p,q).

Principal angles in pseudo-euclidean spaces of index 1

In this paper, we introduce the notion of principal angles between subspaces of the same signature in a (real, complex or quaternionic) pseudo-euclidean space of index 1. We show that these determine the relative position of an important class of pairs of hyperbolic and elliptic subspaces (Theorems 3.9 and 4.10). Also, as an application, we will see that these angles can be used to study the relative position of pairs of an important class of totally geodesic submanifolds of the (real, complex or quaternionic) hyperbolic space (Theorem 6.3). As a consequence we obtain a nice interpretation of these principal angles as geometric invariants of the hyperbolic space.

The scalar curvature of CR submanifold of maximal CR dimension with a conformal change

In this paper, we investigate the codimension reduction theorem for an n-dimensional submanifold of an (n + p)-dimensional manifold (M, ~g) with a conformal change g = exp(–f)g where g denotes the Fubini-study metric on (n + p)-dimensional complex projective space P n+p/2 (C). Moreover, we tend to calculate the scalar curvature of an n-dimensional CR submanifold of maximal CR dimension of (M; g) and achieve the sufficient conditions for the existence of a totally geodesic submanifold M0 that includes M.

Special Lagrangians in nearly Kähler [formula omitted

This article explores special Lagrangian submanifolds in CP3, viewed as a nearly Kähler manifold, from two different perspectives. Intrinsically, using a moving frame set-up, and extrinsically, using SU(2) moment-type maps. We describe new homogeneous examples, from both perspectives, and classify totally geodesic special Lagrangian submanifolds. We show that every special Lagrangian in CP3, or the flag manifold F1,2(C3) admitting a symmetry of an SU(2) subgroup of nearly Kähler automorphisms is automatically homogeneous.

Holomorphic Retractions of Bounded Symmetric Domains onto Totally Geodesic Complex Submanifolds

Holomorphic Retractions of Bounded Symmetric Domains onto Totally Geodesic Complex Submanifolds

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.