arXiv research

We give the expression of the metric derived from Lie groups. For the metric derived from classical Lie groups such as the unitary group, the orthogonal group and the symplectic group, we conjecture that the metric becomes the Einstein metric.

Defines tangent spaces on causal sets using partial derivatives and metrics.

The paper shows objective derivatives are covariant derivatives on Riemannian metrics.

Study uniquely determines Riemannian metric derivatives from boundary data.

We relate Ambrosio-Kirchheim metric currents to Alberti representations and Weaver derivations. In particular, given a metric current $T$, we show that if the module $\mathscr{X}(\|T\|)$ of Weaver derivations is finitely generated, then $T$ can be represented in terms of derivations; this extends previous results of Wi…

Derives spacetime regularity under specific curvature conditions.

Study on geodesics of Finsler metrics derived from Riemannian metrics.

Invariant covariant derivatives on homogeneous spaces are characterized.

Study of time-dependent metrics and connections in geometry.

Develops symmetric Cartan calculus linking to Patterson-Walker metric.

The problem for consistency between linear transports along paths and real bundle metrics in real vector bundles is stated. Necessary and/or sufficient conditions, as well as conditions for existence, for such consistency are derived. All metrics (resp. transports) consistent with a given transport (resp. metric) are e…

Formula derived for volume entropy of certain metrics on Euclidean space.

We investigate the relationship between measurable differentiable structures on doubling metric measure spaces and derivations. We prove: [1] a decomposition theorem for the module of derivations into free modules; [2] the existence of a measurable differentiable structure assuming that one can control the pointwise up…

We derive a precise asymptotic expansion of the complete Kähler-Einstein metric on the punctured Riemann sphere with three or more omitting points. By using Schwarzian derivative, we prove that the coefficients of the expansion are polynomials on the two parameters which are uniquely determined by the omitting points. …

Local fractional derivatives affect Riemann curvature tensor to zero.

Metric functions for phoneme perception capture the similarity structure among phonemes in a given language and therefore play a central role in phonology and psycho-linguistics. Various phenomena depend on phoneme similarity, such as spoken word recognition or serial recall from verbal working memory. This study prese…

Sharp estimates derived for quasilinear equations on metric measure spaces.

This paper, sixth in a series of eight, uses the geometric calculus on manifolds developed in previous papers of the series to introduce through the concept of a metric extensor field g a metric structure for a smooth manifold M. The associated Christoffel operators, a notable decomposition of that object and the assoc…

Derives Kerr metric from two commuting complex structures.

Formula derived for curvature on smooth manifolds.

Geodesic coordinates derived for a specific metric in surface group representations.

For a nonconstant holomorphic map between projective Riemann surfaces with conformal metrics, we consider invariant Schwarzian derivatives and projective Schwarzian derivatives of general virtual order. We show that these two quantities are related by the "Schwarzian derivative" of the metrics of the surfaces (at least…

New derivation of Type IIA flow metrics.

Let $(M,g)$ be a Riemannian manifold, and $m$ be a second metric on $M$. We give expressions of $m$'s associated connection, and Riemann curvature tensor $R_m$, in terms of $R_g$ and certain combinations of covariant derivatives of $m$ (with respect to the Levi-Civita connection associated with $g$). The formulas turn …

New K3 metrics derived from torus orbifold loci.

Derives smooth homogeneous structures for low-rank tensors.

The paper derives inequalities for submanifolds in quaternionic Kaehler manifolds.

New Finsler metrics derived from pedal curves.

Metric learning has attracted a lot of interest over the last decade, but the generalization ability of such methods has not been thoroughly studied. In this paper, we introduce an adaptation of the notion of algorithmic robustness (previously introduced by Xu and Mannor) that can be used to derive generalization bound…

The Chern sectional curvature of a Hermitian manifold is derived and related to Kähler metrics.

In this paper, we derive apriori estimates for constant scalar curvature Kähler metrics on a compact Kähler manifold. We show that higher order derivatives can be estimated in terms of a $C^0$ bound for the Kähler potential. We also discuss some local versions of these estimates which can be of independent interest.

Proves existence of Yamabe metrics on conical manifolds with conical points and links.

We develop a comprehensive geometric framework for defining spaces $\mathcal{G}(M,E)$ of nonlinear generalized sections of vector bundles $E \to M$ containing spaces of distributional sections $\mathcal{D}'(M, E)$. Our theory incorporates classical differential geometric operations (like tensor products, covariant deri…

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

Small neural networks embed arbitrary metric spaces into Gaussian mixtures.

Derives derivatives and geometric framework for functions with non-independent variables.

Derive Dirichlet scalar curvature energy functional variation formula

We propose a new approach for metric learning by framing it as learning a sparse combination of locally discriminative metrics that are inexpensive to generate from the training data. This flexible framework allows us to naturally derive formulations for global, multi-task and local metric learning. The resulting algor…

Michor and Mumford have shown that the distances between planar curves in the simplest metric (not involving derivatives) are identically zero. We consider two conformally equivalent metrics for which the distances between curves are nontrivial. We show that in the case of the simpler of the two metrics, the only minim…

Paper establishes inequality for submanifolds in real space forms with semi-symmetric non-metric connection.

We consider the local solution to the Calabi flow for C^αinitial metric. We also prove that the Calabi flow on compact Kaehler surfaces can be extended once the metrics along the flow are bounded in L^\infty sense. This can be viewed as obtaining higher order derivative estimates from second order derivatives for a fou…

The Gauss-Bonnet curvature of order $2k$ is a generalization to higher dimensions of the Gauss-Bonnet integrand in dimension $2k$, as the usual scalar curvature generalizes the two dimensional Gauss-Bonnet integrand. In this paper, we evaluate the first variation of the integrals of these curvatures seen as functionals…

We give a maximum principle proof of interior derivative estimates for the Kähler-Ricci flow, assuming local uniform bounds on the metric.

We bring together those systems of hydrodynamical type that can be written as geodesic equations on diffeomorphism groups or on extensions of diffeomorphism groups with right invariant $L^2$ or $H^1$ metrics. We present their formal derivation starting from Euler's equation, the first order equation satisfied by the ri…

Researchers compute curvatures of Stiefel manifolds with new metrics.

In this paper we study the extent to which conformally compact asymptotically hyperbolic metrics may be characterized intrinsically. Building on the work of the first author, we prove that decay of sectional curvature to -1 and decay of covariant derivatives of curvature outside an appropriate compact set yield Hölder …

In Finsler geometry the complete lift vector fields have distinguished geometric significance. For example a vector field on a Finsler manifold is said to be conformal if its complete lift is conformal in usual sense. In this work we define a new Riemannian or Pseudo-Riemannian metric on TM derived from a Finsler metri…

The paper constructs Einstein Sasaki metrics on solvable Lie groups.