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.
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.
The paper establishes new Casorati inequalities for various Riemannian maps and submersions.
problem Developing new inequalities for Riemannian maps and submersions.
method Using general forms of Casorati inequalities, the paper derives inequalities for specific Riemannian spaces.
result The paper provides new Casorati inequalities for Riemannian maps and submersions.
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 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.
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. 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.
Paper establishes new inequality for Riemannian maps and applies it to various space forms.
problem Developing a new inequality for Riemannian maps and its applications.
method Proposed and utilized a general Chen's first inequality for Riemannian maps and applied it to various space forms.
result Validated the new inequality and compared results with existing approaches.
Paper studies invariant submanifolds in Sasakian spaces, finding conditions for total geodesy.
problem Investigating invariant submanifolds in Sasakian spaces.
method Analyzing equivalent conditions for total geodesy.
result Obtained conditions for invariant submanifolds to be totally geodesic.
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.
Study invariant submanifolds in generalized Sasakian-space-forms.
problem Characterize invariant submanifolds in generalized Sasakian-space-forms.
method Analyzes invariant submanifolds with respect to Levi-Civita and semi-symmetric metric connections.
result Provides necessary and sufficient conditions for total geodesic submanifolds.
In this paper, we show that a generalized Sasakian space form of dimension greater than three is either of constant sectional curvature; or a canal hypersurface in Euclidean or Minkowski spaces; or locally a certain type of twisted product of a real line and a flat almost Hermitian manifold; or locally a wapred product…
Completes the space of vector-valued one-forms on manifolds.
problem Metric incompleteness of the space of full-ranked one-forms.
method Distance equality and quotient structures.
result Concrete description of the metric completion of the space of full-ranked one-forms.
Paper derives inequalities for submanifolds in a specific geometric space.
problem Chen's inequalities for submanifolds in (κ,μ)-contact space form. method Using generalized semi-symmetric non-metric connections.
result Derives new inequalities for submanifolds.
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.
It is shown that, in a generalized S-space-form, F7 always equals F8.
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.
We show that there is no phi-recurrent generalized Sasakian-space-forms, when is a non-zero constant.
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.
Study of differential forms and vector fields on orbit spaces.
problem Understanding vector fields and differential forms on orbit spaces.
method Defined differential forms and vector fields as multilinear maps on infinitesimal diffeomorphisms.
result Intrinsic view of vector fields and differential forms on orbit spaces.
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.
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.
We present a general definition of the Poisson bracket between differential forms on the extended multiphase space appearing in the geometric formulation of first order classical field theories and, more generally, on exact multisymplectic manifolds. It is well defined for a certain class of differential forms that we …
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.
Extends a proof for a geometric problem in curved spaces.
problem Overdetermined Serrin's problem in curved spaces.
method Introduces a P-function to generalize Weinberger's proof.
result Generalizes a proof for a geometric problem in curved spaces.
The paper proves rigidity theorems for forms on reductive symmetric spaces.
problem Local rigidity of forms on reductive symmetric spaces under representations of discrete groups.
method General local rigidity theorem for pull-backs of homogeneous forms, reinterpretation of old results.
result Volume of closed manifolds is constant under deformation of G/H-structure. The paper derives Chen-Ricci inequalities for Riemannian submersions and maps.
problem Chen-Ricci inequalities for Riemannian submersions and maps.
method General forms of Chen-Ricci inequalities for Riemannian submersions and maps are derived, involving curvatures of subspaces.
result New, easy, and elegant techniques for Chen-Ricci inequalities are established.
We construct a C-space associated with every closed 3-form on a spacetime M and show that it depends on the class of the form in H3(M,Z). We also demonstrate that C-spaces have a relation to generalized geometry and to gerbes. C-spaces are constructed after introducing additional coordinates at the open sets and …
Proves Mayer-Vietoris sequence for diffeological spaces using generating families.
problem No specific problem stated; focuses on extending a sequence.
method Uses generating families instead of coverings in diffeological spaces.
result Proves a Mayer-Vietoris sequence for diffeological spaces.
Defines virtual immersions to characterize symmetric spaces.
problem Characterizing symmetric spaces using virtual immersions.
method Defines virtual immersions as a generalization of isometric immersions, proves characterization of symmetric spaces.
result A manifold admits a virtual immersion with skew symmetric second fundamental form if and only if it is a symmetric space.
The paper defines conditions for para-Kaehler immersions and classifies them between forms.
problem Existence and classification of para-Kaehler immersions.
method Necessary and sufficient conditions for para-Kaehler immersions in para-Kaehler space forms.
result Local para-Kaehler immersions cannot be globally extended on simply connected para-Kaehler manifolds.
Generalized (κ,μ)-space forms are introduced and studied. We examine in depth the contact metric case and present examples for all possible dimensions. We also analyse the trans-Sasakian case.
Study quasi-Einstein metrics on real hypersurfaces of complex space forms.
problem No Einstein metrics in non-flat complex space forms, explore quasi-Einstein condition.
method Analyze real hypersurfaces in complex Euclidean space, classify quasi-Einstein metrics.
result Classify quasi-Einstein metrics on real hypersurfaces of complex Euclidean space.
The study proves umbilical points have positive measure in 3D space forms.
problem Characterizing umbilical points in 3D space forms.
method Analyzing mean curvature and proving measure properties.
result Umbilical points have positive measure in 3D space forms.
Researchers generalize space forms in Riemannian geometry using specific vector fields.
problem Generalizing space forms in Riemannian geometry with a distinguished vector field.
method Proposing and studying pairs (g,T) of Riemannian metrics and vector fields, leading to Lorentzian metrics with constant curvature.
result Only flat, product manifolds with universal covering as a product of R and N are the only pairs (g,T) whose corresponding Lorentzian metric is a space form in the compact setting.
Study shows most symmetric and 3-symmetric spaces lack solvable Clifford-Klein forms.
problem Existence of compact solvable Clifford-Klein forms in homogeneous spaces.
method Combination of Hirzebruch-Kobayashi-Ono proportionality principle and syndetic hull theory.
result Almost all symmetric and 3-symmetric spaces do not admit solvable compact Clifford-Klein forms.
The paper studies biharmonic conformal hypersurfaces in Riemannian manifolds.
problem Characterizing biharmonic conformal hypersurfaces in Riemannian manifolds.
method Deriving biharmonic equations and proving properties of conformal immersions.
result Properties of biharmonic conformal hypersurfaces in space forms.
In this paper we prove two sharp inequalities involving the normalized scalar curvature and the generalized normalized δ-Casorati curvatures for slant submanifolds in quaternionic space forms. We also characterize those submanifolds for which the equality cases hold. These results are a generalization of some recent …
Researchers describe Hodge-Laplace spectrum on lens spaces.
problem Characterizing lens spaces based on their form spectra.
method Explicit description and rational function encoding of spectra.
result Geometric characterization of lens spaces with identical spectra.
Hodge decomposition extended to H1 space on non-compact manifolds.
problem Extending Hodge decomposition to non-compact manifolds and Sobolev spaces.
method Generalization of Hodge decomposition to H1 space on non-compact manifolds of nonpositive curvature. result Hodge decomposition established for H1 space on non-compact manifolds of nonpositive curvature. The generic singularities and bifurcations are classified for one-parameter families of curves with frames in a space form, the Euclidean space, the elliptic space or the hyperbolic space via projective geometry. Two kinds of frames are considered, adapted frames and osculating frames, in terms of certain differential …
The paper classifies and constructs rotational surfaces with constant astigmatism in space forms.
problem Classifying surfaces with constant astigmatism in space forms.
method Classification and construction of surfaces using variational problems and binormal evolution.
result Locally constructed all rotational surfaces of constant astigmatism.
Canonical structure of the space-time symmetric analogue of the Hamiltonian formalism in field theory based on the De Donder-Weyl (DW) theory is studied. In n space-time dimensions the set of n polymomenta is associated to the space-time derivatives of field variables. The polysymplectic (n+1)-form generalizes th…
Hypersurfaces with constant Ricci eigenvalues in real space forms are classified.
problem Classification of curvature homogeneous hypersurfaces in real space forms
method Proving the converse of curvature homogeneity implies constant Ricci eigenvalues
result Hypersurfaces with constant Ricci eigenvalues in real space forms are classified
The study of real hypersurfaces in pseudo-Riemannian complex space forms and para-complex space forms, which are the pseudo-Riemannian generalizations of the complex space forms, is addressed. It is proved that there are no umbilic hypersurfaces, nor real hypersurfaces with parallel shape operator in such spaces. Denot…
In this paper, we give a definition of coherent tangent bundles of space form type, which is a generalized notion of space forms. Then, we classify their realizations in the sphere as a wave front, which is a generalization of a theorem of O'Neill and Stiel: any isometric immersion of the n-sphere into the (n+1)-sphere…
Study calculates curvature functionals in space forms, proving sphere stability.
problem Understanding curvature functionals in space forms.
method Computed first and second variations of a general functional, applying stability criteria.
result Proved spheres are stable in a generalized Willmore functional.
The space of broken hyperbolic structures generalizes the Teichmüller space of a punctured surface, and the space of projectivized broken measured foliations (equivalently, the space of projectivized affine foliations) generalizes the space of projectivized measured foliations. Just as projectivized measured foliations…