arXiv research

To determine the Lie groups that admit a flat (eventually complete) left invariant semi-Riemannian metric is an open and difficult problem. The main aim of this paper is the study of the flatness of left invariant semi Riemannian metrics on quadratic Lie groups i.e. Lie groups endowed with a bi-invariant semi Riemannia…

Counterexample disproves conjecture on flat metrics and fiber bundles.

The paper explores F-manifolds and metrics, constructing canonical structures.

Classifies all flat Riemannian metrics on the plane, including complete and incomplete cases.

Length metrics can be closely approximated by conformally flat metrics.

Study of flat metrics on orbifolds and their moduli spaces.

In these notes we survey basic concepts of affine geometry and their interaction with Riemannian geometry. We give a characterization of affine manifolds which has as counterpart those pseudo-Riemannian manifolds whose Levi-Civita connection is flat. We show that no connected semisimple Lie group admits a left invarian…

We define a class of two dimensional surfaces conformally related to minimal surfaces in flat three dimensional geometries. By the utility of the metrics of such surfaces we give a construction of the metrics of $2 N$ dimensional Ricci flat (pseudo-) Riemannian geometries.

With a f-left-invariant Riemannian metric on a Lie group $G$, we mean a Riemannian metric which is conformally equivalent to a left-invariant Riemannian metric, with the conformal factor $f$. In this article, we study the geometry of such metrics and give a necessary and sufficient condition for an f-left-invariant Rie…

This work proves certain general orbifold compactness results for spaces of Riemannian metrics, generalizing earlier results along these lines for Einstein metrics or metrics with bounded Ricci curvature. This is then applied to prove such compactness for spaces of Bach-flat (for example half-conformally flat) metrics …

Study constructs non-Riemannian Finsler metrics using warped product.

We call the Lie algebra of a Lie group with a left invariant pseudo-Riemannian flat metric pseudo-Riemannian flat Lie algebra. We give a new proof of a classical result of Milnor on Riemannian flat Lie algebras. We reduce the study of Lorentzian flat Lie algebras to those with trivial center or those with degenerate ce…

Solves a problem in Riemannian geometry for scalar-flat metrics with boundary conditions.

Researchers found a non-Ricci-flat Einstein metric on a 7D nilpotent Lie group.

Study describes flat metric moduli spaces on 4D manifolds.

We consider the sigma models where the base metric is proportional to the metric of the configuration space. We show that the corresponding sigma model equation admits a Lax pair. We also show that this type of sigma models in two dimensions are intimately related to the minimal surfaces in a flat pseudo Riemannian 3-s…

The paper classifies special types of contact metric manifolds with curvature conditions.

New equivalence found for flat vector bundles without extra conditions.

New findings on isospectral tori and harmonic maps between flat tori.

Characterizes curvature positivity for Riemannian metrics on flat vector bundles.

In this paper we study a class of Finsler metrics defined by a Riemannian metric and an 1-form. We classify those of projectively flat in dimension $n\geq3$ by a special class of deformations. The results show that the projective flatness of such kind of Finsler metrics always arises from that of some Riemannian metric…

Study relates Finsler structures to Clifford bundles for flat metrics.

Study of pseudo-Riemannian metrics related to Monge-Ampère structures.

Prove Riemannian positive mass theorem for asymptotically flat metrics with low-codimension singularities.

In this paper we prove that every Riemannian metric on a locally conformally flat manifold with umbilic boundary can be conformally deformed to a scalar flat metric having constant mean curvature. This result can be seen as a generalization to higher dimensions of the well known Riemann mapping Theorem in the plane.

New Finsler metrics defined by Riemannian and 1-forms are studied.

The paper proves compactness of scalar-flat metrics on low-dimensional manifolds with umbilic boundary.

We study two types of isotropic planes: weakly isotropic and strongly isotropic planes. We prove that a Riemannian manifold of indefinite metric is conformally flat if and only if its curvature tensor vanishes on all the strongly isotropic planes. We specialize the plane axiom for Riemannian manifolds of indefinite met…

The paper describes flat Hessian metrics on surfaces and their potentials.

Study on flat metrics on 3D and 4D manifolds, focusing on topology and algebra.

New methods find Ricci-flat metrics on specific Lie groups.

The paper studies conformally flat cubic metrics with isotropic curvature, finding they must be Minkowski.

Paper shows limits of Heisenberg manifolds are flat tori.

On a given closed connected manifold of dimension two, or greater, we consider the squared $L^2$-norm of the scalar curvature functional over the space of constant volume Riemannian metrics. We prove that its critical points have constant scalar curvature, and use this to show that a metric is a solution of the critica…

Positive mass theorem for non-smooth metrics on flat manifolds with corners.

Inspired by the Gromov-Hausdorff distance, we define the intrinsic flat distance between oriented $m$ dimensional Riemannian manifolds with boundary by isometrically embedding the manifolds into a common metric space, measuring the flat distance between them and taking an infimum over all isometric embeddings and all c…

We prove that for each closed smooth spin 4-manifold M there exists a closed smooth 4-manifold N such that the connected sum M # N admits a conformally flat Riemannian metric.

We prove that any base space of Riemannian submersion from a compact Lie group (with bi-invariant metric) must have a basic property previously known for normal biquotients; namely, any zero-curvature plane exponentiates to a flat.

The local classification of conformally flat Lorentzian manifolds with special holonomy groups is obtained. The corresponding local metrics are certain extensions of Riemannian spaces of constant sectional curvature to Walker metrics.

Study rigidity in low-regularity Riemannian and semi-Riemannian metrics.

There is a one-to-one correspondence between associated families of generic conformally flat (local-)hypersurfaces in 4-dimensional space forms and conformally flat 3-metrics with the Guichard condition. In this paper, we study the space of conformally flat 3-metrics with the Guichard condition: for a conformally flat …

We determine all Ricci flat left invariant Lorentzian metrics on simply connected 2-step nilpotent Lie groups. We show that the $2k+1$-dimensional Heisenberg Lie group $H_{2k+1}$ carries a Ricci flat left invariant Lorentzian metric if and only if $k=1$. We show also that for any $2\leq q\leq k$, $H_{2k+1}$ carries a R…

New compact examples of pseudo-Kähler manifolds with essential conformal transformations found.

We call a connected Lie group endowed with a left-invariant Lorentzian flat metric Lorentzian flat Lie group. In this Note, we determine all Lorentzian flat Lie groups admitting a timelike left-invariant Killing vector field. We show that these Lie groups are 2-solvable and unimodular and hence geodesically complete. M…

Researchers generalize space forms in Riemannian geometry using specific vector fields.

Notions of compatible and almost compatible pseudo-Riemannian metrics, which are motivated by the theory of compatible (local and nonlocal) Poisson structures of hydrodynamic type and generalize the notion of flat pencil of metrics, are introduced and studied.

The paper studies null hypersurfaces in 4-manifolds with a specific metric structure.

In this paper, we study a special class of Finsler metrics, $(α,β)$-metrics, defined by $F = αφ(\fracβα)$, where $α$ is a Riemannian metric and $β$ is a 1-form. We find an equation that characterizes Ricci-flat $(α,β)$-metrics under the condition that the length of $β$ with respect to $α$ is constant.