Study on Riemannian Poisson warped product spaces and their properties.
problem Characterizing and understanding Riemannian Poisson warped product spaces.
method Formal treatment of Killing and 2-Killing 1-forms on Riemannian Poisson manifolds, including Bochner type results.
result Characterization of 2-Killing 1-form on (R2,g,Π) and Bochner type results on compact spaces. 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.
Linear classifiers in product space forms improve scRNA-seq data classification.
problem Linear classification in products of Euclidean, spherical, and hyperbolic spaces.
method Novel formulations of linear classifiers on Riemannian manifolds, proving expressive power, and formalizing perceptron and SVM classifiers.
result Linear classifiers in product space forms have the same expressive power as in Euclidean space of the same dimension.
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 optimal inequality for contact CR-warped product submanifolds in cosymplectic space forms.
problem Optimal inequality for contact CR-warped product submanifolds in cosymplectic space forms.
method Using Gauss and Codazzi equations, prove an optimal inequality.
result Prove an optimal inequality for contact CR-warped product submanifolds.
Study on real hypersurfaces in products of complex space forms, proving rigidity and nonexistence results.
problem Existence and properties of totally umbilical real hypersurfaces in complex space forms.
method Analyzing shape operators and local product structures in products of complex space forms.
result Nonexistence and rigidity results for totally umbilical real hypersurfaces in products of complex space forms.
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.
Classifies hypersurfaces with constant curvature in product spaces.
problem Classifying hypersurfaces with constant curvature in product spaces.
method Analyzing hypersurfaces in R^k x S^{n-k+1} and R^k x H^{n-k+1} for 2 <= k <= n-1.
result Complete description of hypersurfaces with constant curvature in product spaces of space forms.
We give a complete classification of submanifolds with parallel second fundamental form of a product of two space forms. We also reduce the classification of umbilical submanifolds with dimension m≥3 of a product $\Q_{k_1}^{n_1}\times \Q_{k_2}^{n_2}$ of two space forms whose curvatures satisfy k1+k2=0 to …
The paper calculates curvatures in a specific type of warped product space.
problem Calculating curvatures in contravariant warped product spaces.
method Introduced sectional curvature, defined null, spacelike, timelike 1-forms, and qualar curvature.
result Found sectional curvatures and qualar curvature for specific examples.
The study defines and constructs hypersurfaces in a product of two space forms.
problem Characterizing hypersurfaces in a product of two space forms.
method Explicit construction using parallel families of hypersurfaces and isoparametric hypersurfaces.
result Classification of hypersurfaces with constant mean curvature and constant product angle function.
We obtain a basic inequality involving the Laplacian of the warping function and the squared mean curvature of any warped product isometrically immersed in a Riemannian manifold without assuming any restriction on the Riemann curvature tensor of the ambient manifold. Applying this general theory, we obtain basic inequa…
The study finds inequalities in warped product manifolds linking isoperimetric and curvature integrals.
problem Finding inequalities in warped product manifolds.
method Proving isoperimetric inequalities and relating them to curvature integrals.
result Sharp eigenvalue and geometric inequalities in space forms.
Study classifies Einstein spaces and warped products in weighted geometry.
problem Characterizing geometric structures of weighted Einstein spaces.
method Complete local classification of weighted Einstein spaces with harmonic Weyl tensor.
result Spaces decompose into Einstein or specific warped products.
In this paper we establish a general inequality involving the Laplacian of the warping functions and the squared mean curvature of any doubly warped product isometrically immersed in a Riemannian manifold. Moreover, we obtain some geometric inequalities for C-totally real doubly warped product submanifolds of generaliz…
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.
We translate into the double forms formalism the basic identities of Greub and Greub-Vanstone that were obtained in the mixed exterior algebra. In particular, we introduce a second product in the space of double forms, namely the composition product, which provides this space with a second associative algebra structure…
We give a complete classification of umbilical surfaces of arbitrary codimension of a product Qk1n1×Qk2n2 of space forms whose curvatures satisfy k1+k2=0.
Study minimal Kähler submanifolds in product of space forms.
problem Existence and properties of minimal isometric immersions of Kähler manifolds into product spaces.
method Analyse obstruction conditions and prove existence results for minimal immersions into Sm−1imesR and Hm−1imesR. result Only minimal isometric immersions of Riemannian surfaces into Sm−1imesR and Hm−1imesR exist. We give local, explicit representation formulas for n-dimensional spacelike submanifolds which are marginally trapped in the Minkowski space, the de Sitter and anti de Sitter spaces and the Lorentzian products of the sphere and the hyperbolic space by the real line.
Paper examines stability of quadratic curvature functionals on product Einstein manifolds.
problem Understanding stability of critical points of quadratic curvature functionals on product Einstein manifolds.
method Analyzes Riemannian functionals defined by L2-norms of Ricci, scalar, Weyl, and Riemannian curvatures. result Product of a spherical space form and a compact hyperbolic manifold is unstable for some quadratic functionals if the first eigenvalue of the Laplacian of the hyperbolic manifold is sufficiently small.
The paper proves a new inequality for CR-warped product submanifolds in complex space forms.
problem Proving a new inequality for CR-warped product submanifolds in complex space forms.
method Developed a first Chen inequality for CR-warped product submanifolds in complex space forms.
result The bound is sharp and uniform in the sign of the holomorphic sectional curvature.
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.
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 spectral invariant generalizes analytic torsion for manifolds with geometric product structure.
problem Generalizing analytic torsion for manifolds with specific geometric product structures.
method Defined multi-torsion as a spectral invariant for compact manifolds with a local geometric product structure, proving metric-independence using Stokes' theorem.
result Proved multi-torsion is metric-independent under suitable conditions.
Introduces a new geometric product for differential forms.
problem Constructing a Clifford algebra from differential forms.
method Introduces and revisits the Graf product with a new framework.
result Provides a structure for constructing a Clifford algebra.
We introduce polar metrics on a product manifold, which have product and warped product metrics as special cases. We prove a de Rham-type theorem characterizing Riemannian manifolds that can be locally decomposed as a product manifold endowed with a polar metric. For a product manifold endowed with a polar metric, our …
We provide a local classification of isometric immersions $f\colon L^p\times_ρM^n\to\Q_c^{p+n+k}$ in codimensions k=1,2 of warped products of Riemannian manifolds into space forms, under the assumptions that n≥k+1 and that Np+n=Lp×ρMn has no points with the same constant sectional curvature c as…
The paper classifies isoparametric hypersurfaces in product spaces with different curvatures.
problem Characterizing isoparametric hypersurfaces in product spaces with varying curvatures.
method Analyzing hypersurfaces in product spaces with constant sectional curvatures.
result Classification of isoparametric hypersurfaces in product spaces with different curvatures.
n this paper, we obtain a geometric inequality between the length of the second fundamental form and the length of Lee form in terms of the warping function for a CR-warped product submanifold in a locally conformal Kaehler space form. The equality case is also investigated. Furthermore, the inequality is discussed for…
Currents on Lie groups form a Hopf algebra structure.
problem Understanding algebraic structure of currents on Lie groups.
method Defined Hopf algebra structure on currents using convolution and wedge product.
result Explicit formulas for Hopf algebra operations on currents are derived.
We provide conditions under which an isometric immersion of a (warped) product of manifolds into a space form must be a (warped) product of isometric immersions.
The paper examines stability of Minkowski inequalities in warped product spaces.
problem Stability of Minkowski-type inequalities for hypersurfaces in warped product spaces.
method Established a stability estimate for the norm of the traceless second fundamental form.
result Proved stability of Minkowski inequalities in specific warped product examples.
Estimates mean curvature, scalar curvature, shape operator in warped products.
problem Estimating geometric properties in warped product spaces.
method Local and global upper estimates for curvature and shape operator.
result Results on pseudo-hyperbolic spaces and space forms.
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…
Study on instantons over product manifolds with a codimension-4 form.
problem Characterize dimension reduction for moduli spaces of generalized ASD instantons.
method Integrability results for families of connections, topological criteria, and explicit descriptions of moduli spaces.
result Complete characterization of dimension reduction for moduli spaces of generalized ASD instantons.
The paper derives inequalities for contact CR-warped product submanifolds in cosymplectic space forms.
problem Establishing inequalities for contact CR-warped product submanifolds in cosymplectic space forms.
method Using the Gauss equation and hypotheses for cosymplectic and nearly cosymplectic manifolds, the paper derives inequalities for the norm of the second fundamental form and the shape operator.
result The contact warped product submanifolds in cosymplectic manifolds exhibit a geometric property called D1-minimality, leading to an optimal general inequality. The study extends Lp-spectrum analysis to warped products and Kleinian groups.
problem Extending Lp-spectrum analysis to new types of manifolds. method Generalized to warped products and certain quotients of hyperbolic space.
result Proves the Lp-spectrum contains a parabolic region for specific manifolds. We consider surfaces with parallel mean curvature vector (pmc surfaces) in CPn×R and CHn×R, and, more generally, in cosymplectic space forms. We introduce a holomorphic quadratic differential on such surfaces. This is then used in order to show that the anti-invariant…
The study classifies isoparametric hypersurfaces in product spaces with constant angle function.
problem Classifying isoparametric hypersurfaces in product spaces with specific curvature conditions.
method Proving constant angle function and using it to classify hypersurfaces.
result Classification of isoparametric and homogeneous hypersurfaces in product spaces.
Analytic torsion form constructed for non-commutative spaces.
problem Constructing analytic torsion form on non-commutative spaces.
method Using fiber bundles, flat vector bundles, and crossed product algebras, we define a non-commutative deRham differential form.
result The constructed torsion form appears in a transgression formula and leads to an index formula.
Constructs moduli space for discs mapping to homogeneous spaces.
problem Stable holomorphic discs mapping to homogeneous spaces.
method Orbifold with corners, fibered products, pushforward, pullback of differential forms.
result Moduli space of stable holomorphic discs as an orbifold with corners.
In this paper we introduce a local approach for the study of maximal surfaces immersed into a Lorentzian product space of the form M2×R1, where M2 is a connected Riemannian surface and M2×R1 is endowed with the product Lorentzian metric. Specifically, we establish a local integral inequality for …
The product of smooth valuations on manifolds is described in terms of differential forms, Gelfand transforms and blow-up spaces. It is shown that the product extends partially to generalized valuations and corresponds geometrically to transversal intersections. This result is used to prove a general kinematic formula …
Formula for spacelike submanifolds in warped products.
problem Finding formulas for spacelike submanifolds in semi-Riemannian warped products.
method Extending Simons' type formulas for spacelike submanifolds in semi-Riemannian warped products.
result Compact spacelike hypersurfaces with parallel mean curvature and non-negative sectional curvature are isoparametric hypersurfaces.
The study examines hypersurfaces in warped products and their properties.
problem Characterizing hypersurfaces in warped products satisfying a specific curvature condition.
method Analyzes hypersurfaces in warped products with a Weingarten condition and proves stability results.
result Hypersurfaces in space forms are geodesic spheres under certain curvature conditions.
Recently, M. Atcken studied Contact CR-warped prod- uct submanifolds in cosymplectic space forms and established gen- eral sharp inequalities for CR-warped products in a cosymplectic manifold [1]. In the present paper, we obtain an inequality for the squared norm of the second fundamental form in terms of constant ?φ…
In this paper, we discuss the Weyl problem in warped product space. We obtain the openness, non rigidity and some applications. These results together with the a priori estimates obtained by Lu imply some existence results. Meanwhile we reprove the infinitesimal rigidity in the space forms.