This paper surveys parallel submanifolds in Riemannian and pseudo-Riemannian manifolds.
problem Understanding parallel submanifolds in Riemannian and pseudo-Riemannian manifolds.
method Comprehensive survey of parallel submanifolds.
result Extrinsic invariants of parallel submanifolds do not vary from point to point.
The study characterizes submanifolds in product spaces.
problem Limited studies on pseudo-umbilical submanifolds.
method Using projections from product structure, conditions for submanifolds to be invariant, anti-invariant, or semi-invariant are derived.
result Necessary and sufficient conditions for pseudo-umbilical submanifolds in locally product Riemannian manifolds.
Researchers describe a specific type of submanifolds in Euclidean space.
problem Understanding inhomogeneous almost symmetric submanifolds.
method Completely describing submanifolds as unions of parallel symmetric submanifolds.
result Described inhomogeneous properly embedded almost symmetric submanifolds as unions of symmetric submanifolds.
The paper finds many infinite-dimensional weakly reflective PF submanifolds in Hilbert spaces.
problem Minimal submanifolds in Hilbert spaces with reflective properties.
method Introduced weakly reflective PF submanifolds into Hilbert spaces and showed their existence.
result Existence of infinite-dimensional weakly reflective PF submanifolds in Hilbert spaces.
We introduce the notions of pointwise almost h-slant submanifolds and pointwise almost h-semi-slant submanifolds as a generalization of slant submanifolds, pointwise slant submanifolds, semi-slant submanifolds, and pointwise semi-slant submanifolds. We have characterizations and investigate the integrability of distrib…
Study of warped product pointwise bi-slant submanifolds in Kaehler manifolds.
problem Characterize warped product pointwise bi-slant submanifolds in Kaehler manifolds.
method Extends previous studies on warped product slant submanifolds.
result Extends several important results on warped product slant submanifolds.
The study explores properties of slant and semi-slant submanifolds in metallic Riemannian manifolds.
problem Characterizing and understanding slant and semi-slant submanifolds in metallic Riemannian manifolds.
method Characterizations and integrability conditions for distributions in semi-slant submanifolds.
result Examples of semi-slant submanifolds in metallic and Golden Riemannian manifolds are provided.
Study lightlike submanifolds in indefinite statistical manifolds, finding conditions and curvature expressions.
problem Characterize lightlike submanifolds in indefinite statistical manifolds.
method Analyze conditions for lightlike submanifolds to be lightlike statistical submanifolds, derive statistical sectional curvature, and investigate induced statistical Ricci tensor symmetry.
result Conditions for lightlike submanifolds to be lightlike statistical submanifolds and expressions for statistical sectional curvature and induced Ricci tensor symmetry.
Develops methods for computing conformal invariants of submanifolds.
problem Computing conformal invariants of submanifolds.
method Direct construction of extrinsic ambient space, global invariants of conformally compact minimal submanifolds, introduction of conformal submanifold scalars.
result Derives an explicit Gauss--Bonnet--Chern-type formula and proves a rigidity result.
The paper introduces austere and arid submanifolds in Hilbert spaces.
problem Classifying minimal orbits in hyperpolar PF actions on Hilbert spaces.
method Introducing austere and arid submanifolds into PF submanifolds in Hilbert spaces.
result Examples of infinite dimensional austere and arid PF submanifolds in Hilbert spaces.
Sphere theorems for submanifolds in Kähler and Sasaki spaces.
problem Proving sphere theorems for Lagrangian and Legendrian submanifolds.
method Differentiable and topological sphere theorems for submanifolds in Kähler and Sasaki spaces.
result Proved sphere theorems for Lagrangian and Legendrian submanifolds.
Study of CR-submanifolds in various Lorentzian manifolds.
problem Exploring CR-submanifolds in different Lorentzian structures.
method Analyzing properties and results of CR-submanifolds in LCS, LP-cosymplectic, S, and GKM manifolds.
result Obtained results on totally umbilical and geodesic CR-submanifolds.
Study attached submanifolds in solvmanifolds, generalizing symmetric space results.
problem Exploring submanifolds in non-symmetric spaces with unusual curvature properties.
method Generalizing Tamaru's construction to pseudo-Riemannian scalar products and root spaces.
result Ricci curvature restriction holds for attached submanifolds under specific algebraic conditions.
Study properties of hemi-slant submanifolds in metallic Riemannian manifolds.
problem Characterize hemi-slant submanifolds in metallic Riemannian manifolds.
method Characterizations and integrability conditions for distributions are derived.
result Examples of hemi-slant submanifolds in metallic and Golden Riemannian manifolds are provided.
The paper derives Chen inequalities for statistical submanifolds in cosymplectic manifolds.
problem Deriving Chen inequalities for statistical submanifolds in cosymplectic manifolds.
method Analyzing statistical cosymplectic manifolds and Legendrian submanifolds to derive Chen inequalities.
result Chen inequalities for statistical submanifolds in cosymplectic manifolds and Legendrian submanifolds are derived.
Survey of recent austere submanifold results.
problem Understanding austere submanifolds.
method Review of existing papers [24,25].
result Recent progress in the field of austere submanifolds.
Study on null submanifolds in indefinite complex contact geometry.
problem Geometry of null submanifolds in indefinite complex contact manifolds.
method Analysis of quaternion null submanifolds and distributions on screen submanifolds.
result Quaternion null submanifolds are always totally geodesic.
Study on deforming calibrated submanifolds with boundary constraints.
problem Deforming calibrated submanifolds with boundary constraints in Riemannian manifolds.
method Extends McLean's deformation theory for closed compact submanifolds to include boundaries.
result Results extend McLean's theory to include boundaries, allowing for more flexible submanifold deformations.
Minimal real Kähler submanifolds in codimension 6 are holomorphic.
problem Characterizing real Kähler submanifolds in high codimension.
method Using isometric rigidity and properties of the second fundamental form.
result Minimal real Kähler submanifolds in codimension 6 are holomorphic.
Study on submanifolds in metallic Riemannian manifolds.
problem Existence of specific submanifolds in metallic Riemannian manifolds.
method Analysis of warped product submanifolds in metallic and Golden Riemannian manifolds.
result Existence and properties of various types of warped product submanifolds.
Study various submanifolds in quaternionic skew-Hermitian spaces.
problem Characterize submanifolds in almost quaternionic skew-Hermitian manifolds.
method Construct explicit examples of submanifolds in semisimple quaternionic skew-Hermitian symmetric spaces.
result Explicit examples of submanifolds for each type considered.
Introduces new lightlike submanifolds in indefinite Kaehler manifolds.
problem Characterizing and understanding new types of lightlike submanifolds.
method Definition and analysis of Screen Transversal Cauchy Riemann (STCR) lightlike submanifolds.
result Characterizes STCR lightlike submanifolds as umbrella of other types.
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 PN(C). By means of the foca…
Proves Frankel theorem for generic submanifolds in Sasakian manifolds.
problem Intersection properties of generic submanifolds in Sasakian manifolds.
method Introduces a weaker notion of generic submanifolds and proves Frankel theorem under specific conditions.
result Derives topological information about generic submanifolds in Sasakian space forms.
The paper normalizes Poisson saturation of coregular submanifolds.
problem Normalizing the Poisson saturation of coregular submanifolds.
method Normal form construction and Poisson geometry analysis.
result Local Poisson saturation of coregular submanifolds is an embedded Poisson submanifold with a normal form.
The paper studies λ-submanifolds in Gauss spaces and proves theorems for complete proper ones.
problem Understanding λ-submanifolds in Gauss spaces and their properties. method Using divergence type theorems and Simons' identities, the authors prove theorems for complete proper λ-submanifolds. result Proves halfspace and gap theorems for complete proper λ-submanifolds, generalizing previous results. Study on special null submanifolds in indefinite Sasakian manifolds.
problem Characterizing null submanifolds in indefinite Sasakian manifolds.
method Proving properties of screen conformal null submanifolds and defining a new class.
result Existence of contact screen conformal r-null submanifolds in indefinite Sasakian space forms. Study on submanifolds of Euclidean space, classifying their symmetry types.
problem Classifying symmetry types of submanifolds in Euclidean space.
method Analyzing properties of full irreducible almost symmetric submanifolds and their cohomogeneity.
result Classification of almost symmetric submanifolds into specific types.
In this paper, invariant submanifolds of a generalized Kenmotsu manifold are studied. Necessary and sufficient conditions are given on a submanifold of a generalized Kenmotsu manifold to be an invariant submanifold.In this case, we investigate further properties of invariant submanifolds of \ a generalized Kenmotsu man…
Study on biwarped product submanifolds in Kähler manifolds, proving their existence and characterizing them.
problem Existence and characterization of biwarped product submanifolds in Kähler manifolds.
method Analyzing multiply warped product submanifolds, providing examples, and establishing inequalities.
result Existence and characterization of non-trivial biwarped product submanifolds in Kähler manifolds.
Constructs biharmonic and r-harmonic submanifolds in cohomogeneity one manifolds.
problem Constructing biharmonic and r-harmonic submanifolds. method Using cohomogeneity one manifolds, the normal index of submanifolds is studied, and new examples are provided.
result Constructs metrics on the sphere with biharmonic non-minimal hypersurfaces.
The paper studies a new type of submanifolds in product spaces.
problem Characterizing and understanding warped product pointwise bi-slant submanifolds.
method Introduced and studied warped product pointwise bi-slant submanifolds of locally product Riemannian manifolds.
result Characterization results and non-trivial examples of these submanifolds.
Recent developments on biconservative submanifolds in Riemannian geometry.
problem Characterizing and understanding biconservative submanifolds.
method Analyzing stress-energy tensors and bitension fields.
result Biconservative submanifolds and H-submanifolds coincide in Euclidean spaces. Uniqueness and stability of minimal submanifolds proved.
problem Uniqueness and stability of minimal submanifolds.
method Proved a strong stability condition on minimal submanifolds.
result Existence and convergence of mean curvature flow for minimal submanifolds.
Study on warped product pointwise semi-slant submanifolds in Sasakian and cosymplectic manifolds.
problem Investigate the geometry of pointwise semi-slant submanifolds and their warped products.
method Using the notion of pointwise slant submanifolds, investigate the geometry of pointwise semi-slant submanifolds and their warped products in Sasakian and cosymplectic manifolds.
result Prove the existence of no proper pointwise semi-slant warped product submanifolds other than contact CR-warped products in Sasakian manifolds.
Diffeological submanifolds are a new type of submanifold in manifold theory.
problem Defining and understanding different types of submanifolds in manifold theory.
method Introducing diffeological submanifolds and comparing them with other types of submanifolds.
result A diffeological submanifold can be included in a manifold without being an immersion.
The study of Euclidean submanifolds with incompressible canonical vector fields.
problem Characterizing Euclidean submanifolds with incompressible canonical vector fields.
method Analyzing the canonical vector field properties and conditions for incompressibility.
result Necessary and sufficient conditions for the canonical vector field of a Euclidean submanifold to be incompressible.
The paper characterizes warped product submanifolds in LCS-manifolds.
problem Characterizing warped product submanifolds in LCS-manifolds.
method Analyzing contact CR-warped product and pseudo-slant submanifolds, and constructing examples.
result There do not exist any proper warped product bi-slant submanifolds of LCS-manifolds.
Study characterizes totally geodesic submanifolds in quotient spaces.
problem Characterizing totally geodesic submanifolds in quotient spaces.
method Characterization through totally geodesic submanifolds and holomorphic tangent sequence splitting.
result Characterization of totally geodesic submanifolds in quotient spaces.
Study on warped product submanifolds in Kenmotsu manifolds.
problem Characterizing warped product pointwise bi-slant submanifolds in Kenmotsu manifolds.
method Characterization and inequality analysis of the submanifolds.
result An inequality for warped product pointwise bi-slant submanifolds in Kenmotsu manifolds is derived.
The paper studies deformations of submanifolds using a new algebraic structure.
problem Deformations of submanifolds in geometric contexts.
method Introduces strongly homotopy Lie algebras to govern deformations of submanifolds.
result Deformations of submanifolds form an analytic variety under certain assumptions.
In this paper we study slant submanifolds of Lorentzian almost contact manifolds. We have taken the submanifold as a space like and then defined the slant angle on a submanifold and thus we extended the results of A. Lotta (Slant submanifolds in contact geometry [8]) and M. A. Khan et. al. (Slant submanifolds of Lorent…
The focal submanifolds of isoparametric hypersurfaces in spheres are all minimal Willmore submanifolds, mostly being A-manifolds in the sense of A.Gray but rarely Ricci-parallel (\cite{QTY},\cite{LY},\cite{TY3}). In this paper we study the geometry of the focal submanifolds via Simons formula. We show that …
Study submanifolds in hyperbolic space, focusing on their boundary and Laplace operator.
problem Understanding the geometry and regularity of submanifolds in hyperbolic space.
method Analyzing asymptotic geometry and regularity properties near the ideal boundary, computing essential spectra.
result Computed essential spectra of the Laplace operator on certain submanifolds.
Classifies totally geodesic submanifolds in Hopf-Berger spheres.
problem Classifying totally geodesic submanifolds in Hopf-Berger spheres.
method Investigates Hopf-Berger spheres, a special family of homogeneous spaces diffeomorphic to spheres constructed via Hopf fibrations.
result Discovered intriguing examples of totally geodesic submanifolds, including isometric submanifolds of real projective spaces and non-congruent submanifolds.
Study the geometry of specific submanifolds in Golden Semi-Riemannian manifolds.
problem Investigate the geometry of specific submanifolds in Golden Semi-Riemannian manifolds.
method Investigate the geometry of distributions and induced connections, provide necessary and sufficient conditions for metric connections, and characterize submanifolds.
result Characterization of screen transversal anti-invariant lightlike submanifolds of Golden Semi-Riemannian manifolds.
Study on deformations of singular submanifolds in complex geometry.
problem Deformation theory of conically singular Cayley submanifolds.
method Proved expected dimension of moduli space and compared complex and Cayley deformations.
result Moduli space is smooth for 2D complex submanifolds of Calabi-Yau 4-folds.
Study of mean curvature flow in hyperkähler manifolds leading to complex Lagrangian submanifolds.
problem Understanding the behavior of mean curvature flow in hyperkähler manifolds.
method Definition of twistor energy and analysis of mean curvature flow starting from hyper-Lagrangian submanifolds.
result Mean curvature flow converges to a complex Lagrangian submanifold for sufficiently small twistor energy.