Formulae for non-symmetric connections derived from covariant derivatives.
problem Deriving commutation formulae for non-symmetric affine connections.
method Covariant derivatives of tensors with respect to symmetric and non-symmetric affine connections.
result Formulae for non-symmetric connections derived from covariant derivatives.
Develops symmetric Cartan calculus linking to Patterson-Walker metric.
problem No specific problem stated; focuses on developing a new calculus.
method Symmetric Cartan calculus, using torsion-free affine connections.
result Symmetric Cartan calculus is a complete analogue of classical Cartan calculus.
We prove that a polar foliation of codimension at least three in an irreducible compact symmetric space is hyperpolar, unless the symmetric space has rank one. For reducible symmetric spaces of compact type, we derive decomposition results for polar foliations.
Study on symmetric braid index of ribbon knots, deriving bounds and characterizations.
problem Understanding the symmetric braid index of ribbon knots.
method Defining symmetric braid index, using Khovanov homology, and calculating bounds.
result Existence of knots with symmetric braid index greater than braid index.
We investigate in detail the class of Euclidean affine Kac-Moody symmetric spaces and their orthogonal symmetric affine Kac-Moody algebras (OSAKAs). These spaces are the only class of Kac-Moody symmetric spaces, that is not directly derived from affine Kac-Moody algebras in the classical sense.
We consider generators of algebraic covariant derivative curvature tensors R' which can be constructed by a Young symmetrization of product tensors W*U or U*W, where W and U are covariant tensors of order 2 and 3. W is a symmetric or alternating tensor whereas U belongs to a class of the infinite set S of irreducible s…
We derive a necessary and sufficient condition for the existence of symmetric space structures on quotients of Banach symmetric spaces. Along the way, we investigate the different kinds of reflection subspaces and their Lie triple systems.
Derives a formula for fermion dimensions in spherically symmetric monopole backgrounds.
problem Calculating the dimension of the plane-wave normalizable kernel for massless fermions in spherically symmetric monopole backgrounds.
method Derives a formula for the dimension of the plane-wave normalizable kernel of the Dirac operator for fermions of any representation of SU(N) in the presence of any spherically symmetric monopole background.
result Derives a formula for the dimension of the plane-wave normalizable kernel of the Dirac operator.
The paper derives inequalities for submanifolds in quaternionic Kaehler manifolds.
problem Analyzing submanifolds in quaternionic Kaehler manifolds.
method Established Chen's and generalized Casorati curvature inequalities.
result Derived inequalities for submanifolds in quaternionic Kaehler manifolds.
We discuss locally simply transitive affine actions of Lie groups G on finite-dimensional vector spaces such that the commutator subgroup [G,G] is acting by translations. In other words, we consider left-symmetric algebras satisfying the identity [x,y].z=0. We derive some basic characterizations of such left-symmetric …
We show how the theory of invariant principal bundle connections for reductive homogeneous spaces can be applied to determine the holonomy of generalised Killing spinor covariant derivatives of the form D=∇+Ω in a purely algebraic and algorithmic way, where Ω:TM→Λ∗(TM) is a left-invariant homo…
The paper studies T-tensor of spherically symmetric Finsler metrics and characterizes metrics satisfying the T-condition.
problem Characterizing spherically symmetric Finsler metrics with vanishing T-tensor.
method Deriving a general expression for the T-tensor and characterizing metrics satisfying the T-condition.
result Characterization of spherically symmetric Finsler metrics with vanishing T-tensor.
Derives symmetric and antisymmetric kernels for quantum physics and chemistry applications.
problem Efficiently handling symmetries and antisymmetries in machine learning for quantum physics and chemistry.
method Symmetrizing and antisymmetrizing conventional kernels, analyzing feature space dimensions, proving kernel properties, proposing Slater determinant representation.
result Efficient evaluation of antisymmetric Gaussian kernels even in high-dimensional state spaces, significant reduction in training data size.
Defines semi-symmetric metric connections on differential forms.
problem Analyzing connections on differential forms.
method Defined and studied semi-symmetric metric connections, computed their curvature and Ricci tensors, and analyzed Lie derivatives.
result Derived Gauss-Codazzi-Ricci equations and properties of canonical, Schouten, and Vrancreanu connections.
The study proves strong cosmic censorship violation for spherically symmetric dust clouds.
problem Violation of strong cosmic censorship for spherically symmetric dust clouds.
method Derived an ordinary differential equation for light rays and used it to prove strong cosmic censorship violation.
result Generic violation of strong cosmic censorship for spherically symmetric dust clouds.
Paper establishes inequality for submanifolds in real space forms with semi-symmetric non-metric connection.
problem Deriving a sharp lower bound for Ricci curvature of submanifolds.
method Using semi-symmetric non-metric connection, derive a lower bound for Ricci curvature in terms of mean curvature vector and second fundamental form.
result Established Hineva inequality for submanifolds with semi-symmetric non-metric connection.
Third-order symmetric Lorentzian manifolds, i.e. Lorentzian manifold with zero third derivative of the curvature tensor, are classified. These manifolds are exhausted by a special type of pp-waves, they generalize Cahen-Wallach spaces and second-order symmetric Lorentzian spaces.
The paper defines symmetric brackets for skew-symmetric algebroids with totally skew-symmetric torsion.
problem Defining symmetric brackets for skew-symmetric algebroids.
method Using connections with totally skew-symmetric torsion and pseudo-Riemannian metrics.
result Explicit formula for the Levi-Civita connection and symmetric brackets on almost Hermitian manifolds.
A new method for efficiently computing derivatives of skew-symmetric matrix exponentials.
problem Efficient computation of derivatives for skew-symmetric matrices.
method Characterization of invertibility, construction of nearby logarithm, and efficient implementation.
result Explicit formulae for differentiation and its inverse of skew-symmetric matrix exponentials.
Counterexample found to estimate for skew-symmetric tensors.
problem Estimate for skew-symmetric tensors was claimed and used for classification results.
method Analysis of the estimate in arXiv:2103.15482.
result Counterexample disproves the estimate for skew-symmetric tensors.
The study examines conditions for symmetric and alternating subgroups in mapping class groups of surfaces.
problem Conditions for torsion elements to generate symmetric or alternating subgroups.
method Analyzes mapping class groups of surfaces, derives necessary and sufficient conditions for conjugates of torsion elements to generate symmetric or alternating subgroups.
result Symmetric or alternating subgroups cannot contain irreducible mapping classes and hyperelliptic involutions.
In this paper, we investigate complete curvature-adapted submanifolds with maximal flat section and trivial normal holonomy group in symmetric spaces of compact type or non-compact type under certain condition, and derive the constancy of the principal curvatures of such submanifolds. As its result, we can derive that …
We introduce the notion of weak commensurabilty of arithmetic subgroups and relate it to the length equivalence and isospectrality of locally symmetric spaces. We prove many strong consequences of weak commensurabilty and derive from these many interesting results about isolength and isospectral locally symmetric space…
New exact sequence links cohomology, automorphisms, and extensions of symmetric quandles.
problem Understanding the structure of extensions and automorphisms in symmetric quandles.
method Derived a four-term exact sequence relating 1-cocycles, second cohomology, and automorphisms.
result Obstruction to automorphisms lies in the second cohomology of symmetric quandles.
Killing forms on Riemannian manifolds are differential forms whose covariant derivative is totally skew--symmetric. We show that a compact simply connected symmetric space carries a non--parallel Killing p--form (p≥2) if and only if it isometric to a Riemannian product Sk×N, where Sk is a round sphere…
The study realizes symmetric spaces as cotangent bundles and finds nonnegative curvature examples.
problem Understanding the geometry of symmetric spaces and their associated vector bundles.
method Realizing symmetric spaces as cotangent bundles of flag manifolds and constructing vector bundles.
result Examples of vector bundles over simply connected manifolds with nonnegative curvature but not nonnegative sectional curvature.
Computed formulas for curvature operators and Poincaré polynomials of symmetric spaces.
problem Calculating curvature operators and Poincaré polynomials for symmetric spaces.
method Explicit formulas derived using quantum numbers and eigenvalue analysis.
result Maximum eigenvalue of curvature operators bounded by Einstein constant, with equality for Hermitian spaces.
Lie PCA improves density estimation on symmetric manifolds.
problem Density estimation for symmetric manifolds.
method Spectral method to approximate Lie algebra of symmetry group.
result Improved sample complexity and density estimation on various data sets.
Paper classifies totally symmetric sets in groups and bounds their sizes.
problem Understanding homomorphisms between groups using totally symmetric sets.
method Full classifications and size bounds for totally symmetric sets in various groups.
result Derives restrictions on homomorphisms between certain groups.
The reduction problem of the chiral field equation on symmetric spaces is studied. It is shown that the symmetric chiral field has infinitely many local conservation laws. A recursive formula for these conservation laws is derived and the first associated integral of motion are given explicitly. Furthermore, the Zakhar…
We derive Mok-Siu-Yeung type formulas for horizontal maps from compact contact locally sub-symmetric spaces into strictly pseudoconvex CR manifolds and we obtain some rigidity theorems for the horizontal pseudoharmonic maps.
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.
New derivation shows how a three-factor learning rule is derived from Oja's rule.
problem Deriving a three-factor learning rule from Oja's rule.
method Using frame theory to systematically derive EGHR-PCA from Oja's rule.
result A principled derivation of a biologically plausible learning rule.
The paper studies para-Sasaki-like manifolds with a new metric connection.
problem Investigating new geometric structures on para-Sasaki-like manifolds.
method Deriving relations between connections, analyzing curvature tensors, studying solitons, constructing examples.
result Derived relations and properties of para-Sasaki-like manifolds with the generalized symmetric metric connection.
It is developed the considerations from (S. M. Minčić, [14, 15]) about curvature tensors and pseudotensors for a non-symmetric affine connection space in this paper. How many kinds of covariant derivatives are enough to be defined for complete researching in the field of non-symmetric affine connection spaces is examin…
The n-dimensional Lorentzian manifolds with vanishing second covariant derivative of the Riemann tensor (2-symmetric spacetimes) are characterized and classified. The main result is that either they are locally symmetric or they have a covariantly constant null vector field, in this case defining a subfamily of Brinkma…
An isometric action of a Lie group on a Riemannian manifold is of cohomogeneity one if the corresponding orbit space is one-dimensional. In this article we develop a conceptual approach to the classification of cohomogeneity one actions on Riemannian symmetric spaces of noncompact type in terms of orbit equivalence. As…
The paper studies Stein-Weiss operators on symmetric tensors, extending previous work.
problem Understanding Stein-Weiss operators on symmetric tensors of arbitrary rank.
method Analyzing the decomposition of tensor spaces into irreducible components and computing Weitzenbock formulas.
result Unified framework for second-order Stein-Weiss operators and tools for geometric analysis.
This paper derives radial fields on manifolds of symmetric positive definite matrices.
problem Lack of an expression for radial fields on manifolds of symmetric positive definite matrices.
method Derives an expression for radial fields on manifolds of symmetric positive definite matrices.
result Derives an expression for radial fields on manifolds of symmetric positive definite matrices.
The paper studies a new connection on Riemannian manifolds and finds conditions for symplectic manifolds.
problem Exploring a new quarter-symmetric non-metric connection on Riemannian manifolds.
method Analyzes the properties and relations of the torsion tensor and curvature tensors of the new connection.
result Conditions for a manifold to be symplectic when endowed with the new connection.
New divergence identity for scalar curvature helps prove rigidity of tensors.
problem Proving rigidity of Codazzi tensors under curvature and invariant conditions.
method Derived a divergence identity for a vector field and applied it to tensor rigidity.
result New proof of Tang-Yan theorem on constant eigenvalues for tensors.
Novel neural network approach on hyperbolic and SPD spaces.
problem Developing neural networks on symmetric spaces of noncompact type.
method Unified formulation of distance from a point to a hyperplane.
result Closed-form expression for point-to-hyperplane distance in higher-rank spaces.
As a difference with the positive-definite Riemannian case, in the Lorentzian case there exists proper second-order symmetric spacetimes, i.e., those with vanishing second covariant derivative of the Riemannian tensor (Rλμνρ;α;β=0) which are not locally symmetric (Rλμνρ;α=0). In fact, they lie in the clas…
New deep learning methods solve symmetric PDEs efficiently.
problem Solving nonlinear symmetric PDEs in high dimensions.
method Design of PointNet and DeepSet neural networks.
result DeepSet networks provide more accurate solutions and gradients.
Volume comparison theorem for rank 1 symmetric spaces proved.
problem Volume comparison for symmetric spaces of non-compact type.
method Normalized Ricci--DeTurck flow to analyze volume functional and derive monotonicity properties.
result Volume comparison theorem established for rank 1 symmetric spaces of non-compact type.
Symmetrizes 4d and 3d BPS quivers for Argyres-Douglas theories.
problem Understanding the relationship between 4d and 3d BPS quivers.
method Analyzes geometric backgrounds and uses skein modules to derive quiver partition functions.
result Proves isomorphism between 4d wall-crossing and unlinking of symmetric quivers.
The paper examines Einstein doubly warped product manifolds with a semi-symmetric metric connection.
problem Characterizing Einstein doubly warped product manifolds with a semi-symmetric metric connection.
method Deriving curvature formulas and proving necessary and sufficient conditions for a manifold to be a warped product.
result Obtained results for Einstein doubly warped product manifolds and Einstein-like doubly warped product manifolds.
We show that the space of algebraic covariant derivative curvature tensors R' is generated by Young symmetrized tensor products W*U or U*W, where W and U are covariant tensors of order 2 and 3 whose symmetry classes are irreducible and characterized by the following pairs of partitions: {(2),(3)}, {(2),(2 1)} or {(1 1)…