Shin-ichi Ohta
We consider Hilbert and Funk geometries on a strongly convex domain in the Euclidean space. We show that, with respect to the Lebesgue measure on the domain, Hilbert (resp. Funk) metric has the bounded (resp. constant negative) weighted Ricci curvature. As one of corollaries, these metric measure spaces satisfy the curvature-dimension condition in the sense of Lott, Sturm and Villani.
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In 1929, Paul Funk and Ludwig Berwald gave a characterization of Hilbert geometries from the Finslerian viewpoint. They showed that a smooth Finsler metric in a convex bounded domain of $\mathbb{R}^n$ is the Hilbert geometry in that domain if and only if it is complete, if its geodesics are straight lines and if its flag curvature is equal to -1. The goal of this chapter is to explain these notions in details, to illustrate the relation between Hilbert geometry, Finsler geometry and the calculus of variations, and to prove the Funk-Berwald characterization Theorem.
A timelike space is a Hausdorff topological space equipped with a partial order relation $<$ and a distance function $ρ$ satisfying a collection of axioms including a set of compatibility conditions between the partial order relation and the distance function. The distance function is defined only on a subset of the product of the space with itself that contains the diagonal, namely, $ρ(x,y)$ is defined if and only if $x<y$ or $x=y$. Distances between pairs of distinct points in a triple $x,y,z$, whenever these distances are defined, satisfy the so-called \emph{time inequality}, which is a rev…
The Funk-Finsler structure on the unit disc in the hyperbolic plane
本文在双曲平面多种模型中构造 Funk-Finsler 结构，证明其在 Klein 单位圆盘中为非 Berwald 的 Douglas metric，并通过显式计算多种曲率揭示了其几何性质。
The Funk-Finsler Structure in the Constant Curvature Spaces
本文研究常曲率空间中的Funk-Finsler结构，通过计算多种曲率揭示其几何性质，给出了不同常曲率空间下S-曲率和旗曲率的界值与欧氏情形的恒值结论。
Volume growth of Funk geometry and the flags of polytopes
本文研究Funk几何中球的体积增长，提出并部分验证了与Mahler和旗猜想相关的极小化猜想，并计算了多胞形情形的高阶渐近项，推广了Santaló点概念。
We define the notion of weak Minkowski metric and prove some basic properties of such metrics. We also highlight some of the important analogies between Minkowski geometry and the Funk and Hilbert geometries.
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.
On families of Finsler metrics
本文针对非对称Finsler度量引入两类度量族，系统研究其测地线、完备性及单位球形状等性质，并应用于Funk几何、Hilbert几何和Teichmüller空间中的Thurston型非对称度量。
In this paper, we study the evolving behaviors of the first eigenvalue of Laplace-Beltrami operator under the normalized Ricci flow of model geometries. In every Bianchi class, we estimate the derivative of the eigenvalue. Then we construct monotonic quantities under the Ricci flow and obtain upper and lower bounds for the eigenvalue.
A projective parameter of a geodesic on a Finsler space is defined to be solution of a certain ODE. Using projective parameter and Funk metric, one can construct a projectively invariant intrinsic pseudo-distance on a Finsler space. In the present work, solutions of the projective parameter's ODE are characterized with respect to the sign of parallel Ricci tensor of a Finsler space. It is shown that the pseudo-distance is trivial on complete Finsler spaces of positive semi-definite Ricci tensor and it is a distance on Finsler spaces of parallel negative definite Ricci tensor. These results gen…
This project serves to analyze the behavior of Ricci Flow in five dimensional manifolds. Ricci Flow was introduced by Richard Hamilton in 1982 and was an essential tool in proving the Geometrization and Poincare Conjectures. In general, Ricci Flow is a nonlinear PDE whose solutions are rather difficult to calculate; however, in a homogeneous manifold, the Ricci Flow reduces to an ODE. The behavior of Ricci Flow in two, three, and four dimensional homogenous manifolds has been calculated and is well understood. The work presented here will describe efforts to better understand the behavior of R…
On a convex body in a Euclidean space, we introduce a new variational formulation for its Funk metric, a Finsler metric compatible with the tautological Finsler structure of the convex body. We generalize the metric on Teichmuller spaces with the Weil-Petersson distance function. A set of similarities the resulting metric structure shares with Thurston's asymmetric metric is noted.
Odd-dimensional manifolds with infinitely many different geometries of positive Ricci curvature
针对n≥5的奇数维闭流形，证明其存在无限多个不同的正Ricci曲率几何，即正Ricci曲率度量模空间有无限多个连通分支，丰富了奇数维流形正Ricci曲率几何的研究。
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-space. We define two dimensional surfaces conformally related to the minimal surfaces in flat three dimensional geometries which enable us to give a construction of the metrics of some even dimensional Ricci flat pseudo Riemannian geometries.
Hamilton's Ricci flow (RF) equations were recently expressed in terms of the edge lengths of a d-dimensional piecewise linear (PL) simplicial geometry, for d greater than or equal to 2. The structure of the simplicial Ricci flow (SRF) equations are dimensionally agnostic. These SRF equations were tested numerically and analytically in 3D for simple models and reproduced qualitatively the solution of continuum RF equations including a Type-1 neckpinch singularity. Here we examine a continuum limit of the SRF equations for 3D neck pinch geometries with an arbitrary radial profile. We show that t…
In earlier work, carrying out numerical simulations of the Ricci flow of families of rotationally symmetric geometries on $S3$, we have found strong support for the contention that (at least in the rotationally symmetric case) the Ricci flow for a ``critical'' initial geometry - one which is at the transition point between initial geometries (on $S^3$) whose volume-normalized Ricci flows develop a singular neck pinch, and other initial geometries whose volume normalized Ricci flows converge to a round sphere - evolves into a ``degenerate neck pinch.'' That is, we have seen in this earlier work…
The paper is devoted to the study of the global geometries of harmonic mappings and infinitesimal harmonic transformations and presents their applications to the theory of Ricci solitons.
We study the behavior of the second order Renormalization Group flow on locally homogeneous metrics on closed three-manifolds. In the cases $\mathbb R^3$ and $\text{SO}(3)\times \R$, the flow is qualitatively the same as the Ricci flow. In the cases $\text{H}(3)$ and $\text{H}(2)\times \R$, if the curvature is small, then the flow expands as in the Ricci flow case, while if the curvature is large, then the flow contracts and forms a singularity in finite time. The main focus of the paper is the flow on the $\text{SU}(2)$, $\text{Nil}$, $\text{Sol}$, and $\text{SL}(2,\R)$ 3-geometries, with two…
In this paper we study backward Ricci flow of locally homogeneous geometries of $4$-manifolds which admit compact quotients. We describe the long-term behavior of each class and show that many of the classes exhibit the same behavior near the singular time. In most cases, these manifolds converge to a sub-Riemannian geometry after suitable rescaling.
It is well-known that the Einstein condition on warpedgeometries requires the fibres to be necessarily Einstein. However, exact warped solutions have often been obtained using one- and two-dimensional bases. In this paper, keeping the dimensions and signatures of the base and the fibre independently arbitrary, we obtain all Ricci-flat warped metrics with flat base in closed form and show that the number of free parameters is one less than the dimension of the base. Without any assumptions on the base and fibre geometry, we then show that a warped geometry is flat, i.e, has vanishing Riemann cu…
Projective and Carrollian geometry at time/space-like infinity on projectively compact Ricci flat Einstein manifolds
本文研究射影紧致Ricci平坦Einstein流形在类时/类空无穷远处的强Carrollian几何构造，通过引入基于非有效齐次模型的新型Cartan几何，揭示了其与底层射影结构的联系并证明其普遍性。
Unified synthetic Ricci curvature lower bounds for Riemannian and sub-Riemannian structures
本文提出通过研究度量测度空间，统一黎曼和亚黎曼几何的合成里奇曲率下界理论。
As part of the general investigation of Ricci flow on complete surfaces with finite total curvature, we study this flow for surfaces with asymptotically conical (which includes as a special case asymptotically Euclidean) geometries. After establishing long-time existence, and in particular the fact that the flow preserves the asymptotically conic geometry, we prove that the solution metric $g(t)$ expands at a locally uniform linear rate; moreover, the rescaled family of metrics $t^{-1}g(t)$ exhibits a transition at infinite time inasmuch as it converges locally uniformly to a complete, finite …
We present in this paper a general approach to study the Ricci flow on homogeneous manifolds. Our main tool is a dynamical system defined on a subset H(q,n) of the variety of (q+n)-dimensional Lie algebras, parameterizing the space of all simply connected homogeneous spaces of dimension n with a q-dimensional isotropy, which is proved to be equivalent in a precise sense to the Ricci flow. The approach is useful to better visualize the possible (nonflat) pointed limits of Ricci flow solutions, under diverse rescalings, as well as to determine the type of the possible singularities. Ancient solu…
We study the evolution of wormhole geometries under Ricci flow using numerical methods. Depending on values of initial data parameters, wormhole throats either pinch off or evolve to a monotonically growing state. The transition between these two behaviors exhibits a from of critical phenomena reminiscent of that observed in gravitational collapse. Similar results are obtained for initial data that describe space bubbles attached to asymptotically flat regions. Our numerical methods are applicable to "matter-coupled" Ricci flows derived from conformal invariance in string theory.
We formulate the fractional Ricci flow theory for (pseudo) Riemannian geometries enabled with nonholonomic distributions defining fractional integro-differential structures, for non-integer dimensions. There are constructed fractional analogs of Perelman's functionals and derived the corresponding fractional evolution (Hamilton's) equations. We apply in fractional calculus the nonlinear connection formalism originally elaborated in Finsler geometry and generalizations and recently applied to classical and quantum gravity theories. There are also analyzed the fractional operators for the entrop…
Suppose $(X,ω)$ is a compact Kähler manifold. We introduce and explore the metric geometry of the $L^{p,q}$-Calabi Finsler structure on the space of Kähler metrics $\mathcal H$. After noticing that the $L^{p,q}$-Calabi and $L^{p'}$-Mabuchi path length topologies on $\mathcal H$ do not typically dominate each other, we focus on the finite entropy space $\mathcal E^{Ent}$, contained in the intersection of the $L^p$-Calabi and $L^1$-Mabuchi completions of $\mathcal H$ and find that after a natural strengthening, the $L^p$-Calabi and $L^1$-Mabuchi topologies coincide on $\mathcal E^{Ent}$. As appl…
In this paper, it is elaborated the theory the Ricci flows for manifolds enabled with nonintegrable (nonholonomic) distributions defining nonlinear connection structures. Such manifolds provide a unified geometric arena for nonholonomic Riemannian spaces, Lagrange mechanics, Finsler geometry, and various models of gravity (the Einstein theory and string, or gauge, generalizations). We follow the method of nonholonomic frames with associated nonlinear connection structure and define certain classes of nonholonomic constraints on Riemann manifolds for which various types of generalized Finsler g…
We study some aspects of spherical symmetric dyonic non-supersymmetric black holes in $4d \; N=1$ supergravity coupled to chiral and vector multiplets on Kähler-Ricci solitons. Then, we have a family of dyonic non-supersymmetric black holes deformed with respect to the flow parameter related to the Kähler-Ricci soliton, which possibly controls the nature of black holes, such as their asymptotic and near horizon geometries. Two types of black holes are discussed, namely a family of dyonic Reissner-Nordström-like black holes and Bertotti-Robinson-like black holes where the scalars are freezing a…
Closed manifolds, model geometries, and volume related differentiable invariants
研究通过等距嵌入观察度量，探讨不同条件下闭流形的Kazdan - Warner类型及sigma不变量，利用相关结果确定了具有特定模型几何的流形的KW类型和sigma不变量，如双曲模型流形及其他几种流形的相关性质。
Here (the last paper in a series of four) we end our presentation of the basics of a systematical approach to the differential geometry of a smooth manifold M (supporting a metric field g and a general connection del) which uses the geometric algebras of multivector and extensors (fields) developed in previous papers. The theory of the Riemann and Ricci fields of the the triple (M,del,g)is investigated to for each particular open set U of M through the introduction of a geometric structure on U, i.e., a triple (U,gamma,g) where gamma is a general connection field on U and g is a metric extenso…
We show that near-horizon geometries in the presence of a positive cosmological constant cannot exist with ring topology. In particular, de Sitter black rings with vanishing surface gravity do not exist. Our result relies on a known mathematical theorem which is a straightforward consequence of a type of energy condition for a modified Ricci tensor, similar to the curvature-dimension conditions for the $m$-Bakry-Émery-Ricci tensor.
In this paper we introduce a synthetic notion of Riemannian Ricci bounds from below for metric measure spaces (X,d,m) which is stable under measured Gromov-Hausdorff convergence and rules out Finsler geometries. It can be given in terms of an enforcement of the Lott, Sturm and Villani geodesic convexity condition for the entropy coupled with the linearity of the heat flow. Besides stability, it enjoys the same tensorization, global-to-local and local-to-global properties. In these spaces, that we call RCD(K,\infty) spaces, we prove that the heat flow (which can be equivalently characterized ei…
We prove that given a pseudo-Riemannian conformal structure whose conformal holonomy representation fixes a totally lightlike subspace of arbitrary dimension, there is, wrt. a local metric in the conformal class defined off a singular set, a parallel, totally lightlike distribution on the tangent bundle which contains the image of the Ricci-tensor. This generalizes results obtained for invariant lightlike lines and planes and closes a gap in the understanding of the geometric meaning of reducibly acting conformal holonomy groups. We show how this result naturally applies to the classification …
We present a string inspired 3D Euclidean field theory as the starting point for a modified Ricci flow analysis of the Thurston conjecture. In addition to the metric, the theory contains a dilaton, an antisymmetric tensor field and a Maxwell-Chern Simons field. For constant dilaton, the theory appears to obey a Birkhoff theorem which allows only nine possible classes of solutions, depending on the signs of the parameters in the action. Eight of these correspond to the eight Thurston geometries, while the ninth describes the metric of a squashed three sphere. It therefore appears that one can c…
On Chen-Teo geometries with cosmological constant
研究非零宇宙常数下Chen-Teo几何的Einstein类比，证明解要么是Plebański-Demiański度量，要么具有反自对偶Weyl张量，并详细分析后者，包括构造不同拓扑的引力瞬子。
Static vacuum near horizon geometries are solutions $(M,g,X)$ of a certain quasi-Einstein equation on a closed manifold $M$, where $g$ is a Riemannian metric and $X$ is a closed 1-form. It is known that when the cosmological constant vanishes, there is rigidity: $X$ vanishes and consequently $g$ is Ricci flat. We study this form of rigidity for all signs of the cosmological constant. It has been asserted that this rigidity also holds when the cosmological constant is negative, but we exhibit a counter-example. We show that for negative cosmological constant if $X$ does not vanish identically, …
Machine Learned Calabi-Yau Metrics and Curvature
本文利用神经网络近似光滑与奇异K3曲面及Calabi-Yau三重体上的Ricci-flat度量，并通过持续同调分析高曲率区域聚集行为，验证了Bogomolov--Yau型不等式及拓扑特征关系。
Einstein manifolds with optical geometries of Kerr type
本文分类满足 Kerr 型光学条件的 Ricci 平坦 Lorentz 流形，证明高维不存在、四维分为两大类，并给出与 Kerr 度量及全纯函数相关的显式构造。
In this article we continue the study of the geometry of $k$-D'Atri spaces, $% 1\leq k$ $\leq n-1$ ($n$ denotes the dimension of the manifold)$,$ began by the second author. It is known that $k$-D'Atri spaces, $k\geq 1,$ are related to properties of Jacobi operators $R_{v}$ along geodesics, since she has shown that ${\operatorname{tr}}R_{v}$, ${\operatorname{tr}}R_{v}^{2}$ are invariant under the geodesic flow for any unit tangent vector $v$. Here, assuming that the Riemannian manifold is a D'Atri space, we prove in our main result that ${\operatorname{tr}}R_{v}^{3}$ is also invariant under th…
Kähler landscapes for complex neural network descents and guarantees including a search and destroy of the Calabi-Yau manifold
该研究将复几何与深度学习结合，以凯勒流形为框架分析复杂神经网络下降的损失景观，揭示卡拉比-丘流形相关特性及负曲率对损失景观的破坏，为深度学习理论提供几何分析视角。
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