Paper classifies special slant surfaces with varying curvature.
problem Classifying surfaces with non-constant mean curvature.
method Analyzing special slant surfaces in complex space forms.
result Complete classification of surfaces with non-constant mean curvature.
Study on new types of manifolds derived from 2D space-forms.
problem Characterizing new types of almost contact B-metric manifolds.
method Used two methods for generating B-metric on product manifolds, characterized with curvature properties.
result New manifolds are classified and their curvature properties are studied.
Study on a specific type of Riemannian manifolds constructed from 2D space-forms.
problem Characterizing and understanding new types of Riemannian manifolds.
method Constructed as a product of a real line and a 2-dimensional Riemannian space-form, with metrics derived from cone and hyperbolic extensions.
result Characterized and studied in terms of their curvature properties.
A Lie algebra structure on variation vector fields along an immersed curve in a 2-dimensional real space form is investigated. This Lie algebra particularized to plane curves is the cornerstone in order to define a Hamiltonian structure for plane curve motions. The Hamiltonian form and the integrability of the planar…
Paper solves isoperimetric problem in specific Finsler space.
problem Isoperimetric problem in 2D Finsler space with k=0.
method Using Busemann-Hausdorff area, proved circle maximizes area.
result Circle centered at origin maximizes area in isoperimetric problem.
The study proves a discrete Blaschke theorem for convex polygons in 2-dimensional space forms.
problem Investigating curvature and circumradius constraints for convex polygons in 2-space forms.
method Defining curvature at each vertex and proving a Blaschke-type theorem.
result The circumradius of a convex polygon satisfies a specific inequality related to its vertex curvatures.
Paper studies isoperimetric problem in 2D Finsler space with k=0.
problem Isoperimetric problem in 2D Finsler space with k=0.
method Holmes-Thompson area used to investigate.
result Circle centered at origin achieves maximum area.
The paper studies Steklov eigenvalues in space forms and warped product manifolds, deriving bounds and monotonicity results.
problem Estimating Steklov eigenvalues in space forms and warped product manifolds.
method Monotonicity results for Steklov eigenvalues in geodesic disks and warped product manifolds with non-negative Ricci curvature.
result Sharp bounds and monotonicity results for Steklov eigenvalues on warped product manifolds.
New discrete curves defined in space forms with geometric properties.
problem Defining discrete elastic and constrained elastic curves in space forms.
method Extending discrete Euclidean curvature to space forms and using Bäcklund transformations.
result Discrete elastic and constrained elastic curves are elements of a curve hierarchy.
Complete classification of isoparametric hypersurfaces in product spaces of space forms.
problem Classifying isoparametric hypersurfaces in product spaces of space forms.
method Proving constant product angle function and removing constant principal curvatures condition.
result Complete classification of isoparametric hypersurfaces in product spaces of space forms.
Study on PMC surfaces in complex space forms, linking biconservative and totally real properties.
problem Characterizing PMC surfaces in complex space forms and their properties.
method Analyzing interactions between PMC, totally real, and biconservative properties; proving rigidity and reduction codimension results.
result PMC surfaces in non-flat complex space forms are biconservative if and only if totally real.
Study of evolutoids and involutoids of convex curves in 2D space forms.
problem Characterizing evolutoids and involutoids of convex curves in various 2D space forms.
method Explicit parametrization and analysis of geodesics, singularity theory, and wavefronts.
result Existence and properties of involutoids for convex curves in M−1,0. A vector field s on a Riemannian manifold M is said to be harmonic if there exists a member of a 2-parameter family of generalised Cheeger-Gromoll metrics on TM with respect to which s is a harmonic section. If M is a simply-connected non-flat space form other than the 2-sphere, examples are obtained of conformal vecto…
Sharp inequalities for submanifolds in space forms derived from elliptic operators.
problem Optimal upper bound for eigenvalues of elliptic operators on submanifolds.
method Using a general symmetric, positive definite tensor T, derived an upper bound for the second eigenvalue of elliptic operators.
result Optimal upper bound for eigenvalues given in terms of integration involving tensor properties and normal vector field.
We classify all the surfaces in M2(c1)×M2(c2) for which the tangent space TpM2 makes constant angles with Tp(M2(c1)×{p2}) (or equivalently with Tp({p1}×M2(c2)) for every point p=(p1,p2) of M2. Here M2(c1) and M2(c2) are 2-dimensional space forms, not both flat.…
A submanifold in space forms satisfies the well-known DDVV inequality due to De Smet, Dillen, Verstraelen and Vrancken. The submanifold attaining equality in the DDVV inequality at every point is called Wintgen ideal submanifold. As conformal invariant objects, Wintgen ideal submanifolds are studied in this paper using…
We study graphs of positive extrinsic curvature with a non-removable isolated singularity in 3-dimensional warped product spaces, and describe their behavior at the singularity in several natural situations. We use Monge-Ampère equations to give a classification of the surfaces in 3-dimensional space forms which are em…
Closed finite gap curves are dense in Sobolev spaces of curves in hyperbolic and Euclidean 3-spaces.
problem Density of closed finite gap curves in Sobolev spaces.
method Proof of density in Sobolev spaces for curves in hyperbolic and Euclidean 3-spaces.
result Closed finite gap curves are dense in Sobolev spaces of curves in hyperbolic and Euclidean 3-spaces.
The study of Bonnet surfaces in 4D space forms reveals new conformally invariant properties and characterizes proper Bonnet surfaces.
problem Investigating Bonnet surfaces in 4D space forms with constant mean curvature.
method Analyzing the moduli space of congruence classes of isometric surfaces, studying properties of lines of curvature, and using infinitesimal isometric deformations.
result Isotropic isothermicity characterizes proper Bonnet surfaces and provides conditions for non-existence of Bonnet mates.
In this paper we consider two special classes of constrained Willmore tori in the 3-sphere. The first class is given by the rotation of closed elastic curves in the upper half plane - viewed as the hyperbolic plane - around the x-axis. The second is given as the preimage of closed constrained elastic curves, i.e., elas…
Two Spin^c structures characterize hypersurfaces in product spaces.
problem Characterizing hypersurfaces in product spaces via Spin^c structures.
method Restricting parallel spinor fields to hypersurfaces.
result Totally umbilical hypersurfaces have constant mean curvature.
A new metric is constructed on tangent bundles of Riemannian manifolds.
problem Constructing a scalar flat metric on tangent bundles.
method Developed a scalar flat metric \( G \) in the tangent bundle \( TN \) of a Riemannian manifold \( (N,g) \).
result The metric \( G \) is locally conformally flat or symmetric under specific conditions.
Study biharmonic submanifolds in complex and Sasakian space forms.
problem Characterize biharmonic submanifolds in generalized space forms.
method Analyze biharmonicity conditions and derive curvature estimates for various submanifolds.
result Obtain curvature estimates for different types of submanifolds.
The study classifies totally umbilical surfaces in warped product manifolds.
problem Classifying totally umbilical surfaces in warped product manifolds.
method Analyzing surfaces in warped product M(κ)fimesI and finding first integrals. result Totally umbilical surfaces in M(κ)fimesI are invariant by isometries. Study solves overdetermined problem on compact surfaces.
problem Overdetermined problem on compact Riemannian surfaces.
method Analyzes Steklov eigenvalues and geometric properties.
result Characterizes domains for which the problem is solvable.
New findings show fundamental group is not audible in spherical space forms.
problem Isospectral spherical space forms with non-cyclic fundamental groups.
method Revisited and found new examples of spherical space forms.
result Fundamental group is not audible among spherical space forms.
Paper studies second order symmetric parallel tensors in generalized f.pk-space forms.
problem Exploring properties of second order symmetric parallel tensors in generalized f.pk-space forms.
method Analyzes the properties of second order symmetric parallel tensors and deduces the existence or non-existence of certain tensors and hypersurfaces.
result There does not exist second order skew-symmetric parallel tensor in f.pk-space form. There is no parallel hypersurface in a generalized f.pk-space form but there is semi-parallel hypersurface.
Classifies weakly Einstein submanifolds in space forms satisfying specific equalities.
problem Characterizing submanifolds in space forms with certain geometric properties.
method Classification based on Chen's equality and semisymmetric conditions.
result Classification of weakly Einstein submanifolds in space forms.
Study finds conditions for Legendre curves to be interpolating sesqui-harmonic in Sasakian space forms.
problem Characterizing Legendre curves in Sasakian space forms.
method Analyzes necessary and sufficient conditions for Legendre curves to be interpolating sesqui-harmonic.
result Obtains an example of an interpolating sesqui-harmonic Legendre curve in a Sasakian space form.
We consider conformally flat hypersurfaces in four dimensional space forms with their associated Guichard nets and Lamé's system of equations. We show that the symmetry group of the Lamé's system, satisfying Guichard condition, is given by translations and dilations in the independent variables and dilations in the dep…
Study f-biharmonic and bi-f-harmonic submanifolds in complex and Sasakian space forms.
problem Characterize f-biharmonic and bi-f-harmonic submanifolds in complex and Sasakian space forms.
method Prove necessary and sufficient conditions for f-biharmonicity and bi-f-harmonicity in general and specific cases.
result Obtain non-existence results for f-biharmonic and bi-f-harmonic submanifolds.
Study f-biharmonic maps and submanifolds, proving rigidity and classifications.
problem Characterize and classify f-biharmonic maps and submanifolds.
method Explore applications, use improved equations, and analyze conformal changes.
result Prove rigidity theorems and classifications of f-biharmonic hypersurfaces.
Study of surfaces in space forms using Lie sphere geometry.
problem Investigate channel linear Weingarten surfaces in various space forms.
method Lie sphere geometric approach to uniform treatment of different ambient geometries.
result Any channel linear Weingarten surface is isothermic and a surface of revolution.
The study examines curvature tensors and solitons in Lorentzian trans-Sasakian space forms.
problem Characterizing curvature tensors and solitons in Lorentzian trans-Sasakian space forms.
method Derivation of various curvature tensors and analysis of solitons under specific conditions.
result Conditions for hyperbolic Ricci and conformal Ricci solitons to be η-Einstein and their expansion/steering/shrinking properties. Researchers provide explicit parametrizations for Sasakian space forms.
problem Understanding homogeneous contact Riemannian structures on Sasakian space forms.
method Explicit parametrizations of all homogeneous structures.
result Detailed descriptions of all homogeneous structures in 3D Sasakian space forms.
The article proves a Hineva inequality for various submanifolds in Quaternionic Space forms.
problem Proving a Hineva inequality for submanifolds in Quaternionic Space forms.
method Establishing the Hineva inequality for different types of submanifolds.
result A Hineva inequality has been proven for various submanifolds in Quaternionic Space forms.
Study on submanifolds in generalized Sasakian-space-forms with various connections.
problem Analyzing submanifolds in generalized Sasakian-space-forms with different connections.
method Examines submanifolds in generalized Sasakian-space-forms with semisymmetric metric, non-metric, Schouten-van Kampen, and Tanaka-webster connections.
result Provides results on submanifolds in generalized Sasakian-space-forms with respect to various connections.
New quasi space forms solve Thurston's geometrical space form problem.
problem Solving Thurston's geometrical space form problem.
method Introducing quasi space forms as non-real space forms with specific geometric properties.
result Quasi space forms offer a metrical, local geometrical solution to Thurston's problem.
Expanding submanifold theory in products of multiple space forms.
problem Generalizing tensors and the Bonnet theorem for products of many space forms.
method Generalizing tensors and the Bonnet theorem from products of 2 space forms to products of many space forms.
result Generalization of tensors and the Bonnet theorem to products of many space forms.
Study on slant submanifolds in complex space forms with a new Simons' type formula.
problem Characterizing slant submanifolds in complex space forms.
method Developed an integral formula of Simons' type for Kaehlerian slant submanifolds.
result Proved a main result using the new Simons' type formula.
Study classifies polyharmonic helices in various space forms.
problem Classifying polyharmonic helices in different space forms.
method Derived classification results for polyharmonic helices in space forms.
result Polyharmonic helices of arbitrary order in space forms of negative curvature are geodesics.
Study biharmonic hypersurfaces in Sasakian space form using Tanaka-Webster connection.
problem Exploring biharmonic hypersurfaces in Sasakian space form.
method Using Tanaka-Webster connection to study biharmonic hypersurfaces.
result Developed new insights into biharmonic hypersurfaces in Sasakian space form.
Study curvature inequalities for real hypersurfaces in complex space forms.
problem Understanding curvature properties of real hypersurfaces in complex space forms.
method Established an inequality relating Ricci curvature, mean curvature, and normal curvature; classified hypersurfaces achieving equality.
result Classified real hypersurfaces in two-dimensional non-flat complex space forms achieving equality in curvature inequality.
Sharp inequalities for submersions from Sasakian space forms.
problem Understanding curvature properties in submersions from Sasakian space forms.
method Analyzing Ricci and scalar curvatures for anti-invariant submersions.
result Deriving precise inequalities involving curvature measures.
The paper identifies unique minimal hypersurfaces in space forms.
problem Finding minimal hypersurfaces in space forms.
method Analyzing Simons' equation to identify minimal hypersurfaces.
result Catenoids and Clifford minimal hypersurfaces are the only complete minimal hypersurfaces satisfying Simons' equation in space forms.
By using the geometric concept of PDEs with prescribed curvature representations, we show that the 1+2 dimensional Landau-Lifshitz equation is gauge equivalent to a 1+2 dimensional nonlinear Schrödinger-type system. From the nonlinear Schrödinger-type system, we construct blowing up H3(R2)-solutions to the 1+…
The paper classifies isoparametric hypersurfaces in Randers space forms.
problem Classifying isoparametric hypersurfaces in Randers space forms.
method Anisotropic submanifolds and isoparametric hypersurfaces in Randers space forms (N,F) with (h,W) were studied.
result The isoparametric hypersurfaces in a Randers space form (N,F) are the same as in Riemannian space, despite different isoparametric functions.
In this article we study isometric immersions of nearly Kähler manifolds into a space form (specially Euclidean space) and show that every nearly Kähler submanifold of a space form has a totally umbilic foliation whose leafs are 6-dimensional nearly Kähler manifolds. Moreover using this foliation we show that there is …