arXiv research

Formula for sectional curvatures on matrix groups.

Study on curvature in finitely generated groups, showing positive curvature in specific cases.

Study on curvatures of surfaces in specific Lie groups.

Study finds all 4D Lie groups with harmonic curvature.

The Cheeger-Gromoll theorem is adapted for groups, revealing geometric properties of virtually abelian groups.

New metrics with special curvature properties are shown to be parallel in certain Lie groups.

Geodesics and curvature of semidirect product groups with right invariant metrics are determined. In the special case of an isometric semidirect product, the curvature is shown to be the sum of the curvature of the two groups. A series of examples, like the magnetic extension of a group, are then considered.

Study magnetic curvature on Lie groups, extending Milnor's work.

The study examines discrete curvature notions on Cayley graphs of certain groups.

Study on surfaces in Heisenberg group with constant mean curvature.

Study on prescribing Ricci curvature on compact Lie groups.

There are five unimodular simply connected three dimensional unimodular non abelian Lie groups: the nilpotent Lie group $\mathrm{Nil}$, the special unitary group $\mathrm{SU}(2)$, the universal covering group $\widetilde{\mathrm{PSL}}(2,\mathbb{R})$ of the special linear group, the solvable Lie group $\mathrm{Sol}$ and…

Study shows a specific Carnot group violates a curvature exponent bound.

Investigates solving curvature equations on special Lie groups.

New examples show non-abelian fundamental groups for positive Ricci curvature manifolds.

The study shows that certain manifolds with positive curvature cannot contain specific geometric structures.

Symmetry groups help define solitons in curved spaces.

Flat surfaces in Lie groups with constant curvature are flat.

Survey on open manifolds with nonnegative Ricci curvature and open questions.

The paper calculates curvature limits and proves Gauss-Bonnet theorems in affine and Minkowski groups.

In this paper, we give the sharp upper bound for the number of vertices with positive curvature in a planar graph with nonnegative combinatorial curvature. Based on this, we show that the automorphism group of a planar---possibly infinite---graph with nonnegative combinatorial curvature and positive total curvature is …

Proves a conjecture for 3D Artin groups using new combinatorial curvature.

The study finds optimal curvature pinching in Heintze groups.

Study duality of zero mean curvature surfaces in Heisenberg group.

The paper discusses algorithms for reconstructing curves with given Euclidean or affine curvatures.

Given a closed smooth manifold M which carries a positive scalar curvature metric, one can associate an abelian group P(M) to the space of positive scalar curvature metrics on this manifold. The group of all diffeomorphisms of the manifold naturally acts on P(M). The moduli group of positive scalar curvature metrics is…

Finite simply connected 2-complexes with nonpositive planar curvature are collapsible.

In Heisenberg group, bisectors are spinal spheres with specific curvature.

The study finds non-trivial elements in moduli spaces of curvature metrics.

New bounds on geodesic dimension and curvature exponent in Carnot groups.

Study left invariant Lorentzian metrics on 3D non-unimodular Lie groups.

Study on $t$-graphs with prescribed mean curvature in Heisenberg groups.

Paper studies fundamental groups of certain Ricci solitons.

We describe the exponential map from an infinite-dimensional Lie algebra to an infinite-dimensional group of operators on a Hilbert space. Notions of differential geometry are introduced for these groups. In particular, the Ricci curvature, which is understood as the limit of the Ricci curvature of finite-dimensional g…

This paper describes cyclic Riemannian Lie groups and their curvatures.

Study on curvature exponent of sub-Finsler Heisenberg groups, proving N_min ≥ 5.

The study shows finite measure-preserving isometry groups for certain metric measure spaces.

The paper finds solutions for specific curvature conditions on 5D Lie groups.

The paper classifies specific types of metrics on 5D Lie groups.

Study of curvature flow on complex Lie groups, leading to soliton convergence.

It is well known that the curvature tensor of a pseudo-Riemannian manifold can be decomposed with respect to the pseudo-orthogonal group into the sum of the Weyl conformal curvature tensor, the traceless part of the Ricci tensor and of the scalar curvature. A similar decomposition with respect to the pseudo-unitary gro…

Proves infinite bordism groups for certain manifolds with positive scalar curvature.

Let M be a Riemannian n-manifold with n greater than or equal to 3. For k between 1 and n, we say M has k-positive Ricci curvature if at every point of M the sum of any k eigenvalues of the Ricci curvature is strictly positive. In particular, one positive Ricci curvature is equivalent to positive Ricci curvature and n-…

Introduces Lie group actions in smoothing processes for currents and spaces with curvature.

Study left-invariant Codazzi tensors and harmonic curvature on Lorentzian Lie groups.

Study crystallographic groups for positive scalar curvature conditions.

Study on surface geometry in Lie groups with CR structures.

Study on surfaces in Heisenberg group with constant mean curvature.