arXiv research

Generalizes Riemann's results on flat coordinates for non-symmetric bilinear forms.

The study finds points on surfaces where a tensor is conformal to a metric.

It is shown how one can apply the classification of the holonomy algebras of Lorentzian manifolds to solve some problems. In particular, a new proof to the classification of Lorentzian manifolds with recurrent curvature tensor is given; the classification of two-symmetric Lorentzian manifolds is explained; conformally …

We study the structure group of a canonical algebraic curvature tensor built from a symmetric bilinear form, and show that in most cases it coincides with the isometry group of the symmetric form from which it is built. Our main result is that the structure group of the direct sum of such canonical algebraic curvature …

In this paper, we extend Su-Zhang's Cheeger-Mueller type theorem for symmetric bilinear torsions to manifolds with boundary in the case that the Riemannian metric and the non-degenerate symmetric bilinear form are of product structure near the boundary. Our result also extends Bruening-Ma's Cheeger-Mueller type theorem…

The theory of harmonic symmetric bilinear forms on a Riemannian manifold is an analogue of the theory of harmonic exterior differential forms on this manifold. To show this, we must consider every symmetric bilinear form on a Riemannian manifold as a one-form with values in the cotangent bundle of this manifold. In thi…

The paper explores geometric decompositions for Ricci tensors and their applications.

New metrics defined on SPD matrices link to divergences and curvature.

Extended current algebra on S^3 with new bilinear form and 2-cocycle.

Bilinear MLPs offer a new way to interpret deep learning models without complex nonlinearities.

The space of tensors of metric curvature type on a Euclidean vector space carries a two-parameter family of orthogonally invariant commutative nonassociative multiplications invariant with respect to the symmetric bilinear form determined by the metric. For a particular choice of parameters these algebras recover the p…

We study natural differential operators transforming two tensor fields into a tensor field. First, it is proved that all bilinear operators are of order one, and then we give the full classification of such operators in several concrete situations.

We are interested in approximation of a multivariate function $f(x_1,\dots,x_d)$ by linear combinations of products $u^1(x_1)\cdots u^d(x_d)$ of univariate functions $u^i(x_i)$, $i=1,\dots,d$. In the case $d=2$ it is a classical problem of bilinear approximation. In the case of approximation in the $L_2$ space the bili…

We generalize a theorem of Bismut-Zhang, which extends the Cheeger-Mueller theorem on Ray-Singer torsion and Reidemeister torsion, to the case where the flat vector bundle over a closed manifold carries a nondegenerate symmetric bilinear form. As a consequence, we prove the Burghelea-Haller conjecture which gives an an…

We present structural properties of Lie algebras admitting symmetric, invariant and nondegenerate bilinear forms. We show that these properties are not satisfied by nilradicals of parabolic subalgebras of real split forms of complex simple Lie algebras, neither by 2-step nilpotent Lie algebras associated with graphs, w…

Paper reduces sample complexity for bilinear systems identification to nearly constant.

The classical Cartan's structural equations show in a compact way the relation between a connection and its curvature, and reveals their geometric interpretation in terms of moving frames. In order to study the mathematical properties of singularities, we need to study the geometry of manifolds endowed on the tangent b…

Fine-grained action segmentation in long untrimmed videos is an important task for many applications such as surveillance, robotics, and human-computer interaction. To understand subtle and precise actions within a long time period, second-order information (e.g. feature covariance) or higher is reported to be effectiv…

Neuc-MDS extends MDS for non-Euclidean data.

Let $V$ be a finite dimensional vector space over a field $F$ of characteristic different from 2, and let $Q$ be a nondegenerate, symmetric, bilinear form on $V$. Let $C\ell(V,Q)$ be the Clifford algebra determined by $V$ and $Q$. The bilinear form $Q$ extends in a natural way to a nondegenerate, symmetric, bilinear fo…

Consider a smooth manifold with a smooth metric which changes bilinear type from Riemann to Lorentz on a hypersurface $Σ$ with radical tangent to $Σ$. Two natural bilinear symmetric forms appear there, and we use it to analyze the geometry of $Σ$. We show the way in which these forms control the smooth extensibility ov…

We extend the main result in the previous paper of Zhang and the author relating the Milnor-Turaev torsion with the complex valued analytic torsion to the equivariant case.

New connections found for quaternionic and para-quaternionic structures.

New mathematical proposal for TQFTs using TMF-modules.

The paper studies invariants of 2-surfaces embedded in 3-space.

Defines metrics and Einstein tensors on Riemannian manifolds, proving vanishing for non-commutative two-torus.

The main topic of this paper is two folds. First, we compute the first relative cohomology group of the Lie algebra of smooth vector fields on the projective line, Vect(RP^1), with coefficients in the space of bilinear differential operators that act on tensor densities, D_{λ, ν;μ}, vanishing on the Lie algebra sl(2,R)…

The paper studies quarter-symmetric connections on Hermitian and Kähler manifolds.

Study biderivations in complete Leibniz algebras, extending Lie algebra results.

In this note we prove the following three algebraic facts which have applications in the theory of holonomy groups and homogeneous spaces: Any irreducibly acting connected subgroup $G \subset Gl(n,\rr)$ is closed. Moreover, if $G$ admits an invariant bilinear form of Lorentzian signature, $G$ is maximal, i.e. it is con…

Sprays on Frechet manifolds connect connections and tangent structures.

Defines metric bundles for manifold geometries, unifying various types of metrics.

Study on tautness tensor for Riemannian foliations.

The paper studies T-tensor of spherically symmetric Finsler metrics and characterizes metrics satisfying the T-condition.

Systematic prolongation for Killing two-tensors in symmetric spaces.

Let V be the pseudo-Euclidean vector space of signature (p,q), p>2 and W a module over the even Clifford algebra Cl^0 (V). A homogeneous quaternionic manifold (M,Q) is constructed for any spin(V)-equivariant linear map Π: \wedge^2 W \to V. If the skew symmetric vector valued bilinear form Πis nondegenerate then (M,Q) i…

New neural networks respect symmetries in symmetric tensors, improving efficiency and generalization.

Defines semi-symmetric metric connection on super warped products.

The paper studies para-Sasaki-like manifolds with a new metric connection.

The paper studies a new connection on Riemannian manifolds and finds conditions for symplectic manifolds.

The paper classifies Killing tensor fields on Riemannian symmetric spaces.

Upper bound found for divergence-free Killing 2-tensors on manifolds.

Characterizes symmetric Killing tensors on specific Lie groups.

We study left-invariant symmetric Killing 2-tensors on 2-step nilpotent Lie groups endowed with a left-invariant Riemannian metric, and construct genuine examples, which are not linear combinations of parallel tensors and symmetric products of Killing vector fields.

Paper studies second order symmetric parallel tensors in generalized f.pk-space forms.

Characterizes Lorentzian manifolds with semi-symmetric metric connections.

New tensors help solve magnetic flow integrability.

Via a non degenerate symmetric bilinear form we identify the coadjoint representation with a new representation and so we induce on the orbits a simplectic form. By considering Hamiltonian systems on the orbits we study some features of them and finally find commuting functions under the corresponding Lie-Poisson brack…