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.
Study on CR-lightlike submanifolds in golden semi-Riemannian manifolds.
problem Exploring properties of CR-lightlike submanifolds in golden semi-Riemannian manifolds.
method Definition and investigation of properties of geodesic CR-submanifolds.
result Obtained results for totally geodesic and totally umbilical CR-submanifolds.
In this paper, we derived biharmonic equations for pseudo-Riemannian submanifolds of pseudo-Riemannian manifolds which includes the biharmonic equations for submanifolds of Riemannian manifolds as a special case. As applications, we proved that a pseudo-umbilical biharmonic pseudo-Riemannian submanifold of a pseudo-Rie…
The aim of our paper is to focus on some properties of slant and semi-slant submanifolds of metallic Riemannian manifolds. We give some characterizations for submanifolds to be slant or semi-slant submanifolds in metallic or Golden Riemannian manifolds and we obtain integrability conditions for the distributions involv…
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.
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.
In the present paper we study pseudo-Riemannian submanifolds which have 3-planar geodesic normal sections.We consider W-curves (helices) on pseudo-Riemannian submanifolds. Finally, we give neccessary and sufficient condition for a normal section to be a W-curve on pseudo-Riemannian submanifolds.
The aim of our paper is to focus on some properties of hemi-slant submanifolds in metallic (and Golden) Riemannian manifolds. We give some characterizations for submanifolds to be hemi-slant submanifolds in metallic (or Golden) Riemannian manifolds and we obtain integrability conditions for the distributions involved. …
Study properties of specific submanifolds in metallic Riemannian spaces.
problem Characterize warped product pointwise bi-slant submanifolds.
method Analyze in locally metallic Riemannian manifolds with examples in Euclidean spaces.
result Construct examples of warped product pointwise bi-slant submanifolds.
In this paper, we study slant submanifolds of Riemannian manifolds with Golden structure. A Riemannian manifold (M~,g~,φ) is called a Golden Riemannian manifold if the (1,1) tensor field φ on M~ is a golden structure, that is φ2=φ+I and the metric $\tild…
The paper studies lightlike submanifolds in bronze semi-Riemannian manifolds with specific geometric properties.
problem Characterizing and understanding lightlike submanifolds in bronze semi-Riemannian manifolds.
method Characterization theorems on geodesicity, integrability, and parallelism of distributions.
result No coisotropic, isotropic, or totally proper screen generic lightlike submanifolds exist.
The paper studies variations of σu-curvature for submanifolds in Riemannian manifolds.
problem Understanding the behavior of σu-curvature under variations of submanifolds. method Analyzes the functional of σu-curvature for submanifolds of arbitrary codimension in Riemannian manifolds. result Provides insights into the variational properties of σu-curvature. Survey on biharmonic Riemannian submersions, a dual concept of biharmonic submanifolds.
problem Understanding biharmonic Riemannian submersions as a dual concept.
method Short survey focusing on biharmonic Riemannian submersions.
result Survey provides insights into the dual concept of biharmonic submanifolds.
The paper defines and studies isoparametric submanifolds in Riemannian Hilbert manifolds.
problem Defining and studying isoparametric submanifolds in Riemannian Hilbert manifolds.
method Introducing curvature-invariant submanifolds, regularizable submanifolds, and isoparametric submanifolds; proving the constancy of mean curvatures and independence of shape operators and normal Jacobi operators.
result Proving that certain submanifolds are isoparametric under specific conditions.
The paper classifies product minimal Lagrangian submanifolds in complex space forms.
problem Understanding minimal Lagrangian submanifolds in complex space forms.
method Examining submanifolds as Riemannian products with constant sectional curvature.
result Complete classification of product minimal Lagrangian submanifolds.
The paper classifies submanifolds in pseudo-Riemannian space forms.
problem Characterizing and classifying totally umbilical submanifolds.
method Classification of congruent classes of totally umbilical submanifolds in non-flat pseudo-Riemannian space forms.
result Some moduli spaces of isometric immersions between space forms are non-Hausdorff.
In this paper, we study the existence of proper warped product submanifolds in metallic (or Golden) Riemannian manifolds and we discuss about semi-invariant, semi-slant and, respectively, hemi-slant warped product submanifolds in metallic and Golden Riemannian manifolds. Also, we provide some examples of warped product…
Study characterizes submanifolds in metallic semi-Riemannian manifolds with specific connections.
problem Characterizing submanifolds in metallic semi-Riemannian manifolds.
method Introduces and analyzes invariant and screen semi-invariant lightlike submanifolds with a quarter symmetric non-metric connection.
result Characterizes integrability and parallelism of distributions in these submanifolds.
Sharp Hardy inequalities on Riemannian submanifolds with non-negative curvature.
problem Establishing Hardy inequalities for submanifolds in Riemannian geometry.
method Analyzing distance functions and using Riemannian submanifolds with non-negative curvature.
result Sharp weighted Hardy inequalities valid for compact and non-compact submanifolds, even in compact ambient manifolds.
Study on heat content for submanifolds in sub-Riemannian geometry.
problem Understanding heat content for submanifolds in sub-Riemannian geometry.
method Existence of smooth tubular neighborhood, definition of relative heat content, approximation via smooth neighborhoods, asymptotic expansion analysis.
result Approximation of relative heat content fails to recover the exact expansion.
Study curvature invariants in sub-Riemannian manifolds.
problem Understand curvature invariants in sub-Riemannian geometry.
method Prove geometrical inequalities for submanifolds with orthogonal distributions.
result Inequalities for submanifolds with orthogonal distributions are derived.
New method creates minimal submanifolds using complex-valued eigenfunctions.
problem Creating minimal submanifolds in compact Riemannian manifolds.
method Employing complex-valued eigenfunctions.
result Manufactured minimal submanifolds in compact Riemannian manifolds.
The paper proves a new inequality for submanifolds in Riemannian space forms and applies it to find minimal submanifolds.
problem Finding necessary conditions for minimal submanifolds in Riemannian space forms.
method Proving a first Chen inequality for general warped product submanifolds and applying it to derive conditions for minimality.
result A necessary condition for submanifolds to be minimal in Riemannian space forms.
Every biharmonic Wintgen ideal submanifold in a Riemannian manifold of constant sectional curvature is either minimal or has constant mean curvature.
problem Biharmonic Wintgen ideal submanifolds in Riemannian manifolds of constant sectional curvature
method Show that every biharmonic Wintgen ideal submanifold in a Riemannian manifold of nonpositive constant sectional curvature is minimal and that every biharmonic Wintgen ideal submanifold in a Riemannian manifold of positive constant sectional curvature has constant mean curvature.
result Partial affirmative answers to Chen's conjecture, generalized Chen's conjecture in hyperbolic spaces, and Balmuş-Montaldo-Oniciuc conjecture in spheres within the class of Wintgen ideal submanifolds.
Study on null helices in semi-Riemannian manifolds with special submanifolds.
problem Investigating geometric properties of null helices on totally umbilical submanifolds in 3D semi-Riemannian manifolds.
method Using the null Frenet frame and degenerate metric condition, equations and invariants characterizing null helices are derived.
result Equations and invariants characterizing null helices on totally umbilical submanifolds in 3D semi-Riemannian manifolds are obtained.
The paper derives optimal inequalities for bi-slant submanifolds in metallic Riemannian space forms.
problem Understanding geometric properties of bi-slant submanifolds in metallic Riemannian product space forms.
method Deriving generalized Wintgen inequality, optimal inequalities involving δ-invariants, Ricci curvature, shape operator invariants, and generalized normalized δ-Casorati curvatures.
result Established optimal inequalities for bi-slant submanifolds in metallic Riemannian product space forms.
Study extends reflective submanifold theory to compact homogeneous spaces.
problem Characterize reflective submanifolds in compact isotropy irreducible spaces.
method Extend previous results to infinite-dimensional Hilbert spaces.
result Inverse image of reflective submanifolds is also reflective.
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 …
New complete minimal submanifolds found in specific Riemannian spaces.
problem Finding complete minimal submanifolds in non-compact Riemannian symmetric spaces.
method Constructing multidimensional families of submanifolds via complex-valued eigenfunctions.
result New families of complete minimal submanifolds in SL_n(R)/SO(n), Sp(n,R)/U(n), SO*(2n)/U(n), and SU*(2n)/Sp(n).
New compact minimal submanifolds found in Riemannian symmetric spaces.
problem Finding compact minimal submanifolds in Riemannian symmetric spaces.
method Constructing multi-dimensional families of compact minimal submanifolds via complex-valued eigenfunctions.
result New families of compact minimal submanifolds of codimension two in SU(n)/SO(n), Sp(n)/U(n), SO(2n)/U(n), and SU(2n)/Sp(n). In this thesis, we study extensions of the theory of Riemannian submanifolds in two directions. First, we will show how Riemannian geometry and submanifold theory in particular, can be generalized using the notion of 'Rinehart spaces', and it will be demonstrated how the developed framework unifies some existing and ne…
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.
The main purpose of the present paper is to study the geometry of screen transversal lightlike submanifolds and radical screen transversal lightlike submanifolds and screen transversal anti-invariant lightlike submanifolds of Golden Semi-Riemannian manifolds. We investigate the geometry of distributions and obtain nece…
The aim of our paper is to focus on some properties of submanifolds in Riemannian manifolds endowed with endomorphisms that generalize the Golden Riemannian structure, named metallic Riemannian structures. We focus on the properties of the structure induced on submanifolds, named by us Σ-metallic Riemannian structure…
The Metallic Ratio is fascinating topic that continually generated news ideas. A Riemannian manifold endowed with a Metallic structure will be called a Metallic Riemannian manifold. The main purpose of the present paper is to study the geometry of transversal lightlike submanifolds and radical transversal lightlike sub…
Sampling random points can reveal submanifold topology.
problem Estimating the topology of submanifolds in Riemannian manifolds.
method Sampling random points in a neighborhood of the submanifold.
result Topology of the submanifold can be recovered with high confidence.
The paper proves inequalities for submanifolds in Riemannian manifolds.
problem Proving geometric inequalities for submanifolds in Riemannian manifolds.
method Using Rauch's comparison theorem and first variation formula.
result General Li-Yau inequality applicable in bounded sectional curvature manifolds.
In the first part of this expository article, the most important constructions and classification results concerning totally geodesic submanifolds in Riemannian symmetric spaces are summarized. In the second part, I describe the results of my classification of the totally geodesic submanifolds in the Riemannian symmetr…
In this article we show how holomorphic Riemannian geometry can be used to relate certain submanifolds in one pseudo-Riemannian space to submanifolds with corresponding geometric properties in other spaces. In order to do so, we shall first rephrase and extend some background theory on holomorphic Riemannian manifolds,…
Study rectifying submanifolds with anti-torqued axis in Riemannian manifolds.
problem Characterize submanifolds with anti-torqued axis in Riemannian manifolds.
method Determine necessary and sufficient conditions for anti-torqued vector fields, characterize submanifolds, and derive rectifying submanifolds as warped products.
result Rectifying submanifolds with anti-torqued axis are warped products with specific warping functions.
Classifies totally geodesic submanifolds in specific geometric spaces.
problem Identifying totally geodesic submanifolds in homogeneous nearly Kähler 6-manifolds and their G2-cones.
method Developed new techniques for studying totally geodesic submanifolds in analytic Riemannian manifolds, homogeneous spaces, and Riemannian cones.
result Obtained a classification of totally geodesic submanifolds in homogeneous nearly Kähler 6-manifolds and their G2-cones.
Study harmonic mappings and submanifolds using Bochner technique.
problem Classical theorems in harmonic mappings and submanifolds.
method Generalized Bochner technique.
result New insights into classical theorems.
We give an equivalent description of taut submanifolds of complete Riemannian manifolds as exactly those submanifolds whose normal exponential map has the property that every preimage of a point is a union of submanifolds. It turns out that every taut submanifold is also Z2-taut. We explicitely construct gen…
In the present paper, we study hemi-slant submanifolds of a locally product Riemannian manifold. We prove that the anti-invariant distribution which is involved in the definition of hemi-slant submanifold is integrable and give some applications of this result. We get a necessary and sufficient condition for a proper h…
Defines natural tensors for submanifolds of pseudo-Riemannian manifolds.
problem Characterizing tensors for submanifolds of pseudo-Riemannian manifolds.
method Constructs geodesic normal coordinates and expresses metric coefficients as polynomials in curvature and second fundamental form derivatives.
result Natural tensors are linear combinations of contractions of curvature and second fundamental form derivatives.
Study submanifolds with boundary in Heisenberg groups, proving Stokes' Theorem.
problem Understanding submanifolds with boundary in sub-Riemannian Heisenberg groups.
method Introduced examples and proved Stokes' Theorem involving Rumin's differential forms.
result Stokes' Theorem for submanifolds with boundary in Heisenberg groups.
Upper bounds for eigenvalues on submanifolds in weighted manifolds.
problem Eigenvalue bounds for submanifolds in weighted Riemannian manifolds.
method Proving upper bounds for divergence-type operators and Steklov problems on submanifolds.
result Reilly-type upper bounds for eigenvalues.
In the present paper, we introduce screen transversal lightlike submanifolds of metallic semi-Riemannian manifolds with its subclasses, namely screen transversal anti-invariant, radical screen transversal and isotropic screen transversal lightlike submanifolds, and give an example. We show that there do not exist co-is…