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 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.
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.
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 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.
New findings on hypersurfaces with specific curvature properties in space forms.
problem Characterizing hypersurfaces with almost constant curvature in space forms.
method Analyzing starshaped hypersurfaces with various curvature conditions.
result Closed starshaped hypersurfaces with almost constant mean curvature or higher order mean curvature are close to geodesic spheres.
We consider interpolating sesqui-harmonic Legendre curves in Sasakian space forms. We find the necessary and sufficient conditions for Legendre curves in Sasakian space forms to be interpolating sesqui-harmonic. Finally, we obtain an example for an interpolating sesqui-harmonic Legendre curve in a Sasakian space form.
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 …
Paper studies quaternionic space forms and Riemannian maps inequalities.
problem Investigate DDVV-type inequality for quaternionic space forms.
method Analyze Riemannian maps from quaternionic space forms to manifolds.
result Derived inequality with equality conditions discussed.
Minimal hypersurfaces are the only H-tensional in 4D space forms.
problem Classifying H-tensional hypersurfaces in 4D space forms. method Investigation of H-tensional hypersurfaces in 4-dimensional space forms of constant sectional curvature. result Minimal hypersurfaces are the only H-tensional hypersurfaces in 4D space forms. Study immersions of Sasakian manifolds into Sasakian space forms.
problem Characterize Sasakian manifolds that can be immersed into Sasakian space forms.
method Proved Calabi's rigidity results, characterised homogeneous Sasakian manifolds.
result Characterized homogeneous Sasakian manifolds that can be embedded into the standard sphere.
Characterizes concircular helices in space forms and ruled hypersurfaces.
problem Understanding concircular hypersurfaces and helices in space forms.
method Characterization through differential equations and ruled hypersurfaces.
result Concircular helices are geodesics of concircular surfaces.
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.
2-stein submanifolds in space forms have constant curvature if normal connection is flat or codimension is 2.
problem Characterizing submanifolds with constant curvature in space forms.
method Analyzing submanifolds with flat normal connection or codimension 2.
result 2-stein submanifolds have constant curvature under specified conditions.
The paper proves isoparametric functions on Finsler space forms under specific conditions.
problem Understanding isoparametric functions in Finsler space forms.
method Proving transnormal functions as isoparametric functions and constructing global and local isoparametric functions using the distance function.
result Generalization of Theorem B to Finsler space forms.
Study ruled real hypersurfaces in nonflat complex space forms with constant norm shape operators.
problem Proving nonexistence and characterizing ruled hypersurfaces in nonflat complex space forms.
method Analyzing shape operators with constant squared norm and using biharmonic properties.
result Biharmonic ruled real hypersurfaces are minimal, leading to their classification.
Classifies actions on complex space forms with Lagrangian orbits.
problem Classifying actions on complex space forms with Lagrangian orbits.
method Classifies holomorphic isometric actions on complex space forms.
result Only examples are Lagrangian affine subspace foliations of complex Euclidean spaces and Lagrangian horocycle foliations of complex hyperbolic spaces.
It is proved, that if M is a connected, complete submanifold of a complex space form N and each geodesic of M lies in an 1-dimensional totally geodesic complex submanifold of N, then M is totally geodesic in N and is a real space form or a complex space form.
We investigate proper biharmonic hypersurfaces with at most three distinct principal curvatures in space forms. We obtain the full classification of proper biharmonic hypersurfaces in 4-dimensional space forms.
Study real hypersurfaces in complex space forms for an inequality involving a contact invariant.
problem Understanding real hypersurfaces in complex space forms and their properties.
method Investigating real hypersurfaces that achieve equality in a specific inequality involving a contact invariant.
result Characterized real hypersurfaces in complex space forms achieving the equality in the inequality.
New non-trivial Kaehler-Ricci solitons found in infinite dimensional complex space forms.
problem Non-trivial Kaehler-Ricci solitons in infinite dimensional complex space forms.
method Exhibited families of non trivial radial Kaehler-Ricci solitons in infinite dimensional complex space forms.
result Non-trivial Kaehler-Ricci solitons exist in infinite dimensional complex space forms, contradicting previous finite dimensional results.
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.
The study shows that the second fundamental form is intrinsic under certain conditions in space forms.
problem Understanding the intrinsic nature of the second fundamental form in space forms.
method Proving the intrinsic nature of the normalized second fundamental form A under specific conditions. result The normalized second fundamental form A is intrinsic if σ2k+1(A)eq0 for some k≥1. Study on biharmonic hypersurfaces in spheres and space forms, proving rigidity under scalar curvature condition.
problem Characterizing biharmonic hypersurfaces in space forms.
method Proved a rigidity result and established an integral formula for biharmonic hypersurfaces.
result Rigidity result under a scalar curvature condition for biharmonic hypersurfaces in space forms.
The paper examines conditions for conformal Ricci solitons on generalized (κ,μ)-space forms.
problem Conditions for conformal Ricci solitons on generalized (κ,μ)-space forms. method Derivation of conditions for solitons to be shrinking, steady, or expanding in terms of conformal pressure p.
result Conditions for a Ricci semi-symmetric generalized (κ,μ)-space form to form an Einstein manifold when equipped with a conformal Ricci soliton. Study proves inequalities for eigenvalues of symmetric domains in space forms.
problem Eigenvalue problem for Laplacian on symmetric domains in space forms.
method Symmetry assumptions on domain, proof of inequalities for eigenvalues.
result Proves Szegö-Weinberger type inequalities for first n positive Neumann eigenvalues.
We consider biharmonic submanifolds in both generalized complex and Sasakian space forms. After giving the biharmonicity conditions for submanifolds in these spaces, we study different particular cases for which we obtain curvature estimates. We consider curves, complex and Lagrangian surfaces and hypersurfaces for the…
Study on ball widths and minimal submanifolds in space forms.
problem Understanding widths of balls and minimal submanifolds.
method Analyzing the area of equatorial balls and related bounds for minimal submanifolds.
result Lower bounds for the area of free boundary minimal submanifolds.
Study on biharmonic and biconservative hypersurfaces in space forms.
problem Properties and behavior of biharmonic and biconservative submanifolds.
method Survey of recent results and alternative proof for hypersurfaces.
result Alternative proof for a well-known result using biconservativity.
Geodesic spheres are the only quasicomplete surfaces in 3-space-forms.
problem Classifying quasicomplete surfaces in 3-space-forms.
method Using quasicompleteness as a weaker form of completeness, the global geometry of surfaces is determined.
result Geodesic spheres are the only quasicomplete surfaces of constant extrinsic curvature in 3-space-forms.
The paper characterizes Whitney and contact Whitney spheres in complex and Sasakian space forms.
problem Characterizing spheres in complex and Sasakian space forms.
method Establishing optimal integral inequalities involving Ricci curvature and second fundamental form norms.
result New characterizations of Whitney and contact Whitney spheres in complex and Sasakian space forms.
The paper examines biconservative hypersurfaces with constant curvature in space forms.
problem Characterizing biconservative hypersurfaces with specific curvature properties.
method Analyzing hypersurfaces with four distinct principal curvatures in space forms.
result Every biconservative hypersurface has constant mean and scalar curvature.
Classifies toric dually flat manifolds into complex space forms.
problem Classifying 1D toric dually flat manifolds.
method Using complex space forms and exponential families.
result Toric dually flat manifolds are complex space forms.
Paper generalizes Minkowski inequality for umbilical hypersurfaces with free boundary.
problem Optimal weighted Minkowski inequality for umbilical hypersurfaces with free boundary.
method Generalization of Xia's result to free boundary case in space forms.
result Free boundary version of optimal weighted Minkowski inequality in space forms.
Study on surfaces pinched by curvature in space forms converging under specific conditions.
problem Investigating convergence of surfaces pinched by curvature in space forms.
method Proving convergence theorems for surfaces pinched by normal curvature in 4-dimensional space forms.
result Generalizes Baker-Nguyen's convergence theorem for surfaces pinched by curvature.
The paper classifies PMCV hypersurfaces in non-flat pseudo-Riemannian space forms.
problem Characterizing PMCV hypersurfaces in non-flat pseudo-Riemannian space forms.
method Analyzing the properties of hypersurfaces with at most two distinct principal curvatures.
result PMCV hypersurfaces are either minimal or locally isoparametric.
The object of this paper is to study the invariant submanifolds of Sasakian generalized-Sasakian-space-form. Here, we obtain some equivalent conditions for an invariant submanifold of a Sasakian generalized-Sasakian-space-forms to be totally geodesic.
Study on immersions with flat normal bundle in curved spaces.
problem Behavior of isometric immersions with negative curvature.
method Investigation of second fundamental form growth in space forms.
result Second fundamental form grows exponentially if normal bundle is flat.
In this paper we establish some inequalities concerning the k-Ricci curvature of a slant submanifold in a quaternionic space form. We also obtain obstructions to the existence of quaternionic slant immersions in quaternionic space forms with unfull first normal bundle.
On real hypersurfaces in complex space forms many results are proven. In this paper we generalize some results concerning extrinsic geometry of real hypersurfaces, to CR submanifolds of maximal CR dimension in complex space forms.
The paper studies stable surfaces with constant curvature in 3D space forms.
problem Stability of surfaces with constant extrinsic curvature in space forms.
method Using stability notions for surfaces with constant higher order mean curvature.
result Rigidity results for surfaces with free boundary in geodesic balls or slabs.
Certain basic inequalities between intrinsic and extrinsic invariants for a submanifold in a (k, m)-contact space form are obtained. As applications we get some results for invariant submanifolds in a (k,m)-contact space form.
Computes tube formulas for valuations in complex space forms.
problem Computing values of valuations on complex space forms.
method Develops tube formulas for valuations in complex space forms and generalizes classical formulas.
result Generalizes classical formulas of Weyl, Gray and others.