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1	Ricci Curvature of Finsler Metrics by Warped Product Marcal, Patricia 05 1900 (has links) Indiana University-Purdue University Indianapolis (IUPUI) / In the present work, we consider a class of Finsler metrics using the warped product notion introduced by B. Chen, Z. Shen and L. Zhao (2018), with another “warping”, one that is consistent with the form of metrics modeling static spacetimes and simplified by spherical symmetry over spatial coordinates, which emerged from the Schwarzschild metric in isotropic coordinates. We will give the PDE characterization for the proposed metrics to be Ricci-flat and construct explicit examples. Whenever possible, we describe both positive-definite solutions and solutions with Lorentz signature. For the latter, the 4-dimensional metrics may also be studied as Finsler spacetimes. Warped Product Finsler Metrics Ricci Curvature Ricci flat
2	Ricci Curvature of Finsler Metrics by Warped Product Patricia Marcal (8788193) 01 May 2020 (has links) <div>In the present work, we consider a class of Finsler metrics using the warped product notion introduced by B. Chen, Z. Shen and L. Zhao (2018), with another “warping”, one that is consistent with the form of metrics modeling static spacetimes and simplified by spherical symmetry over spatial coordinates, which emerged from the Schwarzschild metric in isotropic coordinates. We will give the PDE characterization for the proposed metrics to be Ricci-flat and construct explicit examples. Whenever possible, we describe both positive-definite solutions and solutions with Lorentz signature. For the latter, the 4-dimensional metrics may also be studied as Finsler spacetimes.</div> Geometry Algebraic and Differential Geometry Warped Product Finsler Metric Ricci Curvature Ricci flat
3	On Complete Non-compact Ricci-flat Cohomogeneity One Manifolds Zhou, Cong 10 1900 (has links) <p>We present an alternative proof of the existence theorem of B\"ohm using ideas from the study of gradient Ricci solitons on the multiple warped product cohomogeneity one manifolds by Dancer and Wang. We conclude that the complete Ricci-flat metric converges to a Ricci-flat cone. Also, starting from a $4n$-dimensional $\mathbb{H}P^{n}$ base space, we construct numerical Ricci-flat metrics of cohomogeneity one in ($4n+3$) dimensions whose level surfaces are $\mathbb{C}P^{2n+1}$. We show the local Ricci-flat solution is unique (up to homothety). The numerical results suggest that they all converge to Ricci-flat Ziller cone metrics even if $n=2$.</p> / Master of Science (MSc) Ricci-flat cohomogeneity one asymptotically conical Lyapunov function Geometry and Topology Geometry and Topology
4	On an ODE Associated to the Ricci Flow Bhattacharya, Atreyee January 2013 (has links) (PDF) We discuss two topics in this talk. First we study compact Ricci-flat four dimensional manifolds without boundary and obtain point wise restrictions on curvature( not involving global quantities such as volume and diameter) which force the metric to be flat. We obtain the same conclusion for compact Ricci-flat K¨ahler surfaces with similar but weaker restrictions on holomorphic sectional curvature. Next we study the reaction ODE associated to the evolution of the Riemann curvature operator along the Ricci flow. We analyze the behavior of this ODE near algebraic curvature operators of certain special type that includes the Riemann curvature operators of various(locally) symmetric spaces. We explicitly show the existence of some solution curves to the ODE connecting the curvature operators of certain symmetric spaces. Although the results of these two themes are different, the underlying common feature is the reaction ODE which plays an important role in both. Riemannian Manifolds Curvature Ricci Flow Ricci-Flat 4-Manifolds Algebraic Curvature Operators Vector Fields Ricci-Flat Kahler Surfaces Riemannian Curvature Operator Kahler Manifolds Ordinary Differential Equations Differential Geometry Integral Curves Geometry

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