arXiv research

The Bakry-Émery-Ricci tensor is extended and comparison theorems are proven.

The paper extends a splitting theorem for a specific type of tensor in Riemannian geometry.

A new method for higher-order co-occurrences in hypergraphs.

The Bakry-Emery generalized Ricci tensor arises in scalar-tensor gravitation theories in the conformal gauge known as the Jordan frame. Recent results from the mathematics literature show that standard singularity and splitting theorems that hold when an energy condition is applied in general relativity also hold when …

We analyze Lorentzian spacetimes subject to curvature-dimension bounds using the Bakry-Émery-Ricci tensor. We extend the Hawking-Penrose type singularity theorem and the Lorentzian timelike splitting theorem to synthetic dimensions $N\le 1$, including all negative synthetic dimensions. The rigidity of the timelike spli…

In this paper, we describe the space of adapted connections on a metric contact manifold through the space of their torsion tensors. The torsion tensor is an element of the space of TM-valued two-forms, which splits into various subspaces. We study the parts of the torsion tensor according to this splitting to complete…

Bordered Floer homology assigns invariants to 3-manifolds with boundary, such that the Heegaard Floer homology of a closed 3-manifold, split into two pieces, can be recovered as a tensor product of the bordered invariants of the pieces. We construct cornered Floer homology invariants of 3-manifolds with codimension-2 c…

The paper extends Einstein condition to 4-manifolds using Hodge splittings.

We present in this paper the formalism for the splitting of a four-dimensional Lorentzian manifold by a set of time-like integral curves. Introducing the geometrical tensors characterizing the local spatial frames induced by the congruence (namely, the spatial metric tensor, the extrinsic curvature tensor and the Riema…

Study submanifolds with relative nullity in space forms using splitting tensor.

Co-Clustering, the problem of simultaneously identifying clusters across multiple aspects of a data set, is a natural generalization of clustering to higher-order structured data. Recent convex formulations of bi-clustering and tensor co-clustering, which shrink estimated centroids together using a convex fusion penalt…

Based on a general $(1+3)$ threading of the spacetime $(M,g)$, we obtain a new and simple splitting of a both the Einstein field equations (EFE) and the conservation laws in $(M,g)$. As an application we obtain the splitting of (EFE) in an almost FLRW universe with energy-momentum tensor of a perfect fluid. In particul…

Batch-splitting (data-parallelism) is the dominant distributed Deep Neural Network (DNN) training strategy, due to its universal applicability and its amenability to Single-Program-Multiple-Data (SPMD) programming. However, batch-splitting suffers from problems including the inability to train very large models (due to…

We consider the Hawking-Penrose singularity theorems and the Lorentzian splitting theorem under the weaker curvature condition of nonnegative Bakry-Emery-Ricci curvature $Ric_f^m$ in timelike directions. We prove that they still hold when $m$ is finite, and when $m$ is infinite, they hold under the additional assumptio…

We study Finsler spacetimes and Killing vector fields taking care of the fact that the generalized metric tensor associated to the Lorentz-Finsler function $L$ is in general well defined only on a subset of the slit tangent bundle. We then introduce a new class of Finsler spacetimes endowed with a timelike Killing vect…

We call a metric quasi-Einstein if the $m$-Bakry-Emery Ricci tensor is a constant multiple of the metric tensor. This is a generalization of Einstein metrics, which contains gradient Ricci solitons and is also closely related to the construction of the warped product Einstein metrics. We study properties of quasi-Einst…

In this paper, we study and partially classify those Riemannian man-ifolds carrying a non-identically vanishing function f whose Hessian is minus f times the Ricci-tensor of the manifold.

We propose an algorithm for the non-negative factorization of an occurrence tensor built from heterogeneous networks. We use l0 norm to model sparse errors over discrete values (occurrences), and use decomposed factors to model the embedded groups of nodes. An efficient splitting method is developed to optimize the non…

For a complete Riemannian manifold $M$ with an (1,1)-elliptic Codazzi self-adjoint tensor field $A$ on it, we use the divergence type operator ${L_A}(u): = div(A\nabla u)$ and an extension of the Ricci tensor to extend some major comparison theorems in Riemannian geometry. In fact we extend theorems like mean curvature…

Probabilistic Temporal Tensor Factorization (PTTF) is an effective algorithm to model the temporal tensor data. It leverages a time constraint to capture the evolving properties of tensor data. Nowadays the exploding dataset demands a large scale PTTF analysis, and a parallel solution is critical to accommodate the tre…

The study classifies Riemannian manifolds with curvature nullity.

We study Lorentzian manifolds with a weight function such that the $N$-Bakry-Émery tensor is bounded below. Such spacetimes arise in the physics of scalar-tensor gravitation theories, including Brans-Dicke theory, theories with Kaluza-Klein dimensional reduction, and low-energy approximations to string theory. In the "…

Let $M$ be an odd-dimensional Euclidean space endowed with a contact 1-form $α$. We investigate the space of symmetric contravariant tensor fields on $M$ as a module over the Lie algebra of contact vector fields, i.e. over the Lie subalgebra made up by those vector fields that preserve the contact structure. If we cons…

Paper introduces a new cosmological volume function and its properties.

We first extend Cheeger-Colding Almost Splitting Theorem to smooth metric measure spaces. Arguments utilizing this extension of the Almost Splitting Theorem show that if a smooth metric measure space has almost nonnegative Bakry-Emery Ricci curvature and a lower bound on volume, then its fundamental group is almost abe…

The study examines conditions for weak nearly cosymplectic manifolds to split into products.

Decomposes $SL_3$ skein algebras for surfaces.

A new approach for efficient data compression in split DNN computing.

Researchers describe local properties of Haantjes operators.

It is proved that a compact Kahler manifold whose Ricci tensor has two distinct, constant, non-negative eigenvalues is locally the product of two Kahler-Einstein manifolds. A stronger result is established for the case of Kahler surfaces. Irreducible Kahler manifolds with two distinct, constant eigenvalues of the Ricci…

The paper proves Laplacian comparison theorems for modified m-Bakry-Emery Ricci tensors on Riemannian manifolds.

In 1996/7, J. Bernstein observed that smooth or analytic supermanifolds that mathematicians study are real or (almost) complex ones, while Minkowski superspaces are completely different objects. They are what we call almost real-complex supermanifolds, i.e., real supermanifolds with a non-integrable distribution, the c…

Paper introduces online tensor inference for real-time data analysis.

A classification of 2-dimensional surfaces imbedded in spacetime is presented, according to the algebraic properties of their shape tensor. The classification has five levels, and provides among other things a refinement of the concepts of trapped, umbilical and extremal surfaces, which split into several different cla…

We obtain integral formulas for a metric-affine space equipped with two complementary orthogonal distributions. The integrand depends on the Ricci and mixed scalar curvatures and invariants of the second fundamental forms and integrability tensors of the distributions. The formulas under some conditions yield splitting…

Compact Kähler spaces with zero first Chern class have special geometric properties.

Study geodesic X-ray transforms on hyperbolic surfaces, proposing new reconstruction methods.

This paper uses Karcher's formulation [Kar99] of the O'Neill tensors [O'N66,Gra67] to derive a concise formula for the family $Ω^ε$ of curvature forms obtained by shrinking the fibers of a submersion $π:M\to B$ of semi-Riemannian manifolds by a factor of $1-ε$. The formula clearly shows that as $ε$ approaches 1, $Ω^ε$ …

We consider spacetimes consisting of a manifold with Lorentzian metric and a weight function or scalar field. These spacetimes admit a Bakry-Émery-Ricci tensor which is a natural generalization of the Ricci tensor. We impose an energy condition on the Bakry-Émery-Ricci tensor and obtain singularity theorems of a cosmol…

Efficient method for tensor linear form inference with noisy incomplete data.

This paper studies the (small) quantum homology and cohomology of fibrations $p: P\to S^2$ whose structural group is the group of Hamiltonian symplectomorphisms of the fiber $(M,\om)$. It gives a proof that the rational cohomology splits additively as the vector space tensor product $H^*(M)\otimes H^*(S^2)$, and invest…

The paper explores generalized Riemannian manifolds with weak metric structures.

Refined asymptotics of scalar-flat ALE four-manifolds

We construct a Hennings type logarithmic invariant for restricted quantum $\mathfrak{sl}(2)$ at a $2\mathsf{p}$-th root of unity. This quantum group $U$ is not braided, but factorizable. The invariant is defined for a pair: a 3-manifold $M$ and a colored link $L$ inside $M$. The link $L$ is split into two parts colored…

Study of SO(3)-irreducible geometry in complex 5D and ternary Pauli exclusion principle.

Proposes FACT, a diagnostic for understanding group fairness trade-offs.

New theorem on flat tori stability using harmonic maps and Ricci flow.

Let $(M^n, g)$ be a compact Kähler manifold with nonpositive bisectional curvature. We show that a finite cover is biholomorphic and isometric to a flat torus bundle over a compact Kähler manifold $N^k$ with $c_1 < 0$. This confirms a conjecture of Yau. As a corollary, for any compact Kähler manifold with nonpositive b…