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.
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.
Characterizes metallic structures on product manifolds and provides conditions for warped products to be metallic.
problem Understanding metallic structures on product manifolds and conditions for warped products to be metallic.
method Characterization through metallic maps and necessary/sufficient conditions for warped products.
result Characterizes metallic structures and provides conditions for warped products to be metallic.
Local Sasaki-Ricci solitons are η-Einstein in certain fiber products of homogeneous Sasakian manifolds.
problem Characterizing local immersions of Sasaki-Ricci solitons into homogeneous Sasakian manifolds.
method Analyzing local Sasakian immersions of Sasaki-Ricci solitons into fiber products of homogeneous Sasakian manifolds.
result Sasaki-Ricci solitons are η-Einstein in certain fiber products of homogeneous Sasakian manifolds. The paper studies geometric properties of a specific type of submanifolds in Kaehler manifolds.
problem Analyzing the geometry of pointwise semi-slant warped products in locally conformal Kaehler manifolds.
method The study extends Chen's inequality for CR-warped product submanifolds and investigates equality cases.
result Several results extend Chen's inequality for CR-warped product submanifolds in Kaehler manifolds.
Nonexistence theorem for product type manifolds, proving no locally conformally flat metrics.
problem Proving nonexistence of locally conformally flat metrics on product manifolds.
method Nonexistence theorem for product type manifolds.
result 4-manifold Σ_g×Σ_h does not admit locally conformally flat metrics for g≥2 and h≥1.
Study properties of bi-warped product submanifolds in specific geometric spaces.
problem Characterize geometric properties of bi-warped product submanifolds.
method Analyze squared norm of second fundamental form and warping functions.
result Relationship between squared norm and warping functions for proper slant submanifolds.
Study on 3D manifolds with specific tensor structures and their properties.
problem Characterizing 3D Riemannian manifolds with tensor structures.
method Investigation of locally conformal Riemannian product manifolds and their associated structures.
result Conditions for additional structures to be parallel and properties of almost Einstein and Einstein manifolds.
Adapted metrics found on complex manifolds.
problem Finding metrics suitable for complex manifolds.
method Characterizing adapted metrics as critical points of a functional.
result Gauduchon metric is adapted on locally conformally product manifolds.
Using as an underlying manifold an alpha-Sasakian manifold we introduce warped product Kaehler manifolds. We prove that if the underlying manifold is an alpha-Sasakian space form, then the corresponding Kaehler manifold is of quasi-constant holomorphic sectional curvatures with special distribution. Conversely, we prov…
Products of LCK manifolds do not admit LCK structures.
problem Whether products of compact complex manifolds admit LCK metrics.
method Classifying known LCK manifolds and proving non-existence of LCK structures in product cases.
result Products of LCK manifolds do not admit LCK structures.
We study some similarities between almost product Riemannian structures and almost Hermitian structures. Inspired by the similarities, we prove lower eigenvalue estimates for the Dirac operator on compact Riemannian spin manifolds with locally product structures. We also provide some examples (limiting manifolds) for t…
We introduce warped product skew semi-invariant submanifolds of order 1 of a locally product Riemannian manifold. We give a necessary and sufficient condition for skew semi-invariant submanifold of order 1 to be a locally warped product. We also prove that the invariant distribution which is involved in the definitio…
Killing tensors on reducible spaces are reducible, except for special cases.
problem Characterizing Killing tensors on reducible spaces.
method Analyzing Killing tensors on product manifolds and their lifts.
result Killing tensors on product manifolds are reducible, except for specific cases.
The paper characterizes biharmonic submersions from product manifolds.
problem Characterizing biharmonic Riemannian submersions from M2imesR. method Local characterizations and by-products of biharmonic submersions.
result Local characterizations of biharmonic Riemannian submersions and uniqueness of a specific submersion.
Study on properties of tangential hypersurfaces in product-like manifolds.
problem Investigating properties of tangential hypersurfaces in product-like manifolds.
method Analyzing basic properties and computing curvature tensor relations.
result Computed relations involving the Riemannian curvature tensor of tangential hypersurfaces.
Study characterizes Einstein metrics in warped product spaces.
problem Characterizing Einstein metrics in warped product spaces.
method Local characterizations and global restatements of known results.
result Restated global characterizations of Einstein manifolds.
Study on metrics with harmonic curvature and maximum warped-product decompositions.
problem Understanding metrics with harmonic curvature and their decompositions.
method Analyzing Riemannian manifolds with specific curvature properties and decompositions.
result Local-isometry types form a finite-dimensional moduli space, and specific metrics are realized.
Complex structures found on product of Sasakian manifolds.
problem Exploring complex structures on the product of two Sasakian manifolds.
method Examined a family of complex structures on the product of two compact Sasakian manifolds, computed Dolbeault cohomology groups.
result None of the complex structures in the family admit compatible locally conformally Kähler metrics if both Sasakian manifolds are of dimension greater than 1.
Locally conformally product structures defined on compact manifolds.
problem Defining and characterizing locally conformally product structures on compact manifolds.
method Using reducible, non-flat, incomplete Riemannian metrics and connections with specific properties.
result Construction of new LCP structures from existing ones and links to number field theory.
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…
No conformal product structures on compact manifolds with constant curvature.
problem Existence of conformal product structures on compact manifolds with constant curvature.
method Analyzing non-flat manifolds and irreducible, compact locally symmetric spaces of non-positive curvature.
result Compact non-flat manifolds with constant sectional curvature admit no conformal product structure.
We characterize Osserman and conformally Osserman Riemannian manifolds with the local structure of a warped product. By means of this approach we analyze the twisted product structure and obtain, as a consequence, that the only Osserman manifolds which can be written as a twisted product are those of constant curvature…
Study of local structure of generalized contact bundles.
problem Little is known about the local structure of generalized contact bundles.
method Proved a local splitting theorem similar to those in Poisson geometry.
result In a neighborhood of a regular point, a generalized contact bundle is either the product of a contact and a complex manifold or the product of a symplectic manifold and a manifold with an integrable complex structure.
We obtain the natural diagonal almost product and locally product structures on the total space of the cotangent bundle of a Riemannian manifold. We find the Riemannian almost product (locally product) and the (almost) para-Hermitian cotangent bundles of natural diagonal lift type. We prove the characterization theorem…
We show that locally conformally flat quasi-Einstein manifolds are globally conformally equivalent to a space form or locally isometric to a pp-wave or a warped product.
The paper studies submanifolds in Hadamard and product manifolds, finding eigenvalue bounds.
problem Eigenvalue estimates for submanifolds in specific geometric settings.
method Investigation of submanifolds with bounded mean curvature in Hadamard manifolds and product manifolds NimesR. result Lower bounds for the weighted fundamental tone and the first eigenvalue of the p-Laplacian. 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 …
The study classifies product affine hyperspheres in R^n+1.
problem Locally strongly convex affine hyperspheres in R^(n+1).
method Classification based on product structure and constant sectional curvatures.
result Complete classification of product affine hyperspheres established.
Let f:M→M be an expansive homeomorphism with dense topologically hyperbolic periodic points, M a compact manifold. Then there is a local product structure in an open and dense subset of M. Moreover, if some topologically hyperbolic periodic point has codimension one, then this local product structure is …
Locally conformally product Lie algebras are characterized and constructed.
problem Characterizing and constructing compact locally conformally product Lie algebras.
method Characterization via closed 1-forms and non-unimodular Lie algebras acting on abelian ones.
result Explicit examples of compact LCP manifolds that are not solvmanifolds.
Study inverse curvature flows with local constraints in warped manifolds.
problem Understanding geometric flows with local constraints in complex manifolds.
method Inverse curvature flows in warped product manifolds with local constraints.
result Longtime existence and smooth convergence to coordinate slices.
We prove conformal versions of the local decomposition theorems of de Rham and Hiepko of a Riemannian manifold as a Riemannian or a warped product of Riemannian manifolds. Namely, we give necessary and sufficient conditions for a Riemannian manifold to be locally conformal to either a Riemannian or a warped product. We…
Backward propagation rules for warped products under Ricci flow.
problem Understanding how warped product structures behave under Ricci flow.
method Establishing sufficient conditions for backward propagation of warped product structures.
result Asymptotically conical shrinkers are multiply-warped products over Einstein manifolds.
Characterizes structures on generalized tangent bundles and CRF-structures.
problem Understanding structures on generalized tangent bundles and CRF-structures.
method Equivalent characterizations and spinor formalism for CRF-structures.
result Characterization of generalized complex manifolds as products and infinitesimal deformations of CRF-structures.
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…
We study the global property of local holomorphic isometric mappings from a class of Kahler manifolds into a product of projective algebraic manifolds with induced Fubini-Study metrics, where isometric factors are allowed to be negative.
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.
We consider closed manifolds that admit a metric locally isometric to a product of symmetric planes. For such manifolds, we prove that the Euler characteristic is an obstruction to the existence of flat structures, confirming an old conjecture proved by Milnor in dimension 2. In particular, the Chern conjecture follows…
In this paper, we show that the simplicial volume of Q-rank one locally symmetric spaces covered by the product of R-rank one symmetric spaces is strictly positive.
Electrostatic systems with specific tensors are locally conformally flat.
problem Understanding the geometry of electrostatic systems with special tensors.
method Proving local conformal flatness for electrostatic manifolds with divergence-free Bach tensor.
result Three-dimensional electrostatic manifolds with divergence-free Bach tensor are locally conformally flat.
On the product of two Finsler manifolds M1 M2, we consider the twisted metric F which is construct by using Finsler metrics F1 and F2 on the manifolds M1 and M2, respectively. We introduce horizontal and vertical distributions on twisted product Finsler manifold and study Creducible and semi-C-reducible properties of t…
This note describes sharp Milnor--Wood inequalities for the Euler number of flat oriented vector bundles over closed Riemannian manifolds locally isometric to products of hyperbolic planes. One consequence is that such manifolds do not admit an affine structure, confirming Chern--Sullivan's conjecture in this case. The…
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 …
In this paper we prove that any complete locally conformally flat quasi-Einstein manifold of dimension n≥3 is locally a warped product with (n−1)-dimensional fibers of constant curvature. This result includes also the case of locally conformally flat gradient Ricci solitons.
The study examines non-negative curvature and conullity of curvature tensors.
problem Analyzing conullity conditions for curvature tensors and their compatibility with non-negative sectional curvature.
method Examined conullity two and three in specific dimensions, and used manifold properties to derive conditions.
result Finite volume manifolds with conullity 3 are locally products, and conditions are compatible with non-negative curvature only for specific manifold types.
Study conformal product structures on reducible Riemannian manifolds.
problem Characterize conformal product structures on compact reducible Riemannian manifolds.
method Analyzing technical assumptions and properties of conformal product structures.
result Under suitable conditions, underlying manifolds are either conformally flat or triple products.
Localized signal representation on graph bundles using Fourier analysis.
problem Representing signals on graph bundles with twists.
method Partition of unity and product factorization over the base graph.
result Lifted bases for signal spaces of graph bundle components.