在這篇畢業論文里，我們將闡述微分球面定理。這是由Brelldle和孫理察在2007年用里奇流所證明的。球面定理的研究在微分幾何中有很長的歷史。人們研究一種稱為δ-pillched的截面曲率條件(δ在0和1 之間) ，使得一個緊致單連通的黎曼流形滿足這個曲率條件就會同胚或者微分同胚于一個球面。里奇流是由哈密爾頓在1982年所引進的，當時，他證明了任意一個閉單連通三維黎曼流形只要滿足正的里奇曲率條件就微分同胚于一個球面。 / 在這里，我們會闡述關於里奇流的一些基本結果，包括曲率在里奇流下變化的方式、短時間存在性、唯一性以及曲率和一般張量的高階導數估計。同時，我們也會講述一般的在里奇流下的極大值原理以及收斂性定理。非負迷向曲率條件能夠在里奇流下得到保持的結果也會被詳細證明。最後，我們通過由Böhm 和Wilkillg斤引進的一簇不變圓錐來完成整個微分球面定涅的證明。 / In this thesis, we present the proof of the differentiable sphere theorem which was proved by Brendle and Schoen in 2007 using the Ricci flow. The sphere theorem in differential geometry has a long history. People studied the δ-pinched sectional curvature condition showing that for various cases of δ ∈ (0, 1), any simply-connected, compact Riemannian manifold with this curvature condition is homeomorphic or diffeomorphic to a sphere. The Ricci flow was introduced by Hamilton in his seminal paper in 1982 which proved that every simply-connected, compact three-manifold with positive Ricci curvature is diffeomorphic to a sphere. / In this text, we present some background materials for the Ricci flow, including curvature evolution under the Ricci flow, short-time existence, uniqueness and higher derivatives estimate for curvature and tensor. We also focus on the maximum principle and convergence criterion for the Ricci flow. The fact thatnonnegative isotropic curvature is preserved under the Ricci flow will be showed.Finally, we complete the proof of the differentiable sphere theorem using a family of invariant cones which was constructed by Böhm and Wilking. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Huang, Shaochuang. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 96-98). / Abstracts also in Chinese. / Abstract --- p.i / Acknowledgements --- p.iii / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Background Materials for The Ricci Flow --- p.6 / Chapter 2.1 --- Evolution of curvature under the Ricci flow --- p.6 / Chapter 2.2 --- Short-time existence --- p.13 / Chapter 2.2.1 --- The Ricci tensor is not elliptic --- p.13 / Chapter 2.2.2 --- DeTurck’s trick --- p.17 / Chapter 2.3 --- Uniqueness --- p.20 / Chapter 2.4 --- Estimates for derivatives of curvature --- p.22 / Chapter 2.5 --- Estimates for derivatives of tensor --- p.25 / Chapter 3 --- Hamilton’s Maximum Principle for The Ricci Flow --- p.30 / Chapter 4 --- Hamilton’s Convergence Criterion for The Ricci Flow --- p.39 / Chapter 5 --- Nonnegative Isotropic Curvature --- p.54 / Chapter 5.1 --- Nonnegative isotropic curvature is preserved --- p.54 / Chapter 5.2 --- The Cone Ĉ --- p.67 / Chapter 6 --- Proof of The Differentiable Sphere Theorem --- p.73 / Chapter 6.1 --- An algebraic identity for curvature tensors --- p.73 / Chapter 6.2 --- Construction of a family of invariant cones --- p.80 / Chapter 6.3 --- Proof of The Differentiable Sphere Theorem --- p.87 / Bibliography --- p.96

Thesis (M.A.)--Columbia International University, 1996. / Typescript. Includes bibliographical references (leaves 85-87).

Thesis (M.A.)--University of Hong Kong, 1966. / Also available in print.

Thesis (M.Phil.)--Hong Kong University of Science and Technology, 2007. / Includes bibliographical references (leaves 50-51). Also available in electronic version.

CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / O objetivo deste trabalho à estudar a geometria dos sÃlitons de Ricci compactos, os quais correspondem as soluÃÃes auto-similires do fluxo de Ricci. AlÃm disso, essas variedades podem ser vistas como uma generalizaÃÃo das mÃtricas de Einstein. Neste trabalho, mostraremos que todo sÃliton de Ricci compacto tem curvatura escalar positiva. Alem disso, mostraremos que o seu grupo fundamental à sempre finito. Em particular, apresentaremos uma prova feita por Perelman [19] que todo sÃliton de Ricci compacto à do tipo gradiente / The aim of this work is to study the geometry of the compact Ricci soliton, which correspond to self-similar solution of the Ricci flow. These manifolds are natural generalization to Einstein metrics. Here we shall prove that every compact Ricci soliton has positive scalar curvature. Moreover, we show that its fundamental group is finite. Finally, we prove that every compact Ricci soliton must be gradient.

BATISTA, Rondinelle Marcolino. Rigidez e estimativas de volume de métricas tipo Einstein. 2016. 66f. Tese (doutorado) - Universidade Federal do Ceará, Centro de Ciências, Programa de Pós-Graduação em Matemática, Fortaleza-Ce, 2016 / Submitted by Rocilda Sales (rocilda@ufc.br) on 2016-11-11T13:44:42Z No. of bitstreams: 1 2016_tese_rmbatista.pdf: 535478 bytes, checksum: 7358fcbe5f72a8a139933cc257dfa85e (MD5) / Approved for entry into archive by Rocilda Sales (rocilda@ufc.br) on 2016-11-11T13:45:23Z (GMT) No. of bitstreams: 1 2016_tese_rmbatista.pdf: 535478 bytes, checksum: 7358fcbe5f72a8a139933cc257dfa85e (MD5) / Made available in DSpace on 2016-11-11T13:45:23Z (GMT). No. of bitstreams: 1 2016_tese_rmbatista.pdf: 535478 bytes, checksum: 7358fcbe5f72a8a139933cc257dfa85e (MD5) Previous issue date: 2016 / The purpose of this work is to study like-Einstein metrics, namely, Ricci solitons, almost Ricci solitons and quasi-Einstein metrics. First, we deduce two compactness theorem for gradient Ricci solitons satisfying certain special conditions. In the sequel we prove some integral formulae which allow us to prove that every compact almost Ricci solitons with constant scalar curvature must be gradient type. Moreover, we prove that every compact locally conformally at gradient Ricci soliton must be isometric to standard sphere under an integral condition. Finally, we study the growth of the geodesic balls of steady quasi-Einstein metrics. Moreover, we use Einstein quasi-metric theory to prove a triviality theorem and then to produce a certain class of Einstein warped products under a suitable hypothesis in the fiber. / Nosso objetivo nesta tese é abordar uma classe de métricas tipo Einstein, a saber sólitons de Ricci, quase sólitons de Ricci e métricas quasi-Einstein. Primeiramente obteremos dois resultados sobre compacidade de sólitons de Ricci gradiente, supondo que o quadrado da norma do campo que define tal sóliton é integrável e a derivada da função curvatura escalar na direção do gradiente da função potencial é não negativa, ou uma certa limitação inferior da função potencial. Em seguida, provaremos algumas fórmulas integrais para quase sóliton de Ricci compacto, que nos permite provar que todo quase sóliton de Ricci compacto com curvatura escalar constante é gradiente. Além disso, mostraremos que todo quase sóliton de Ricci gradiente localmente conformemente plano é isométrico a esfera euclidiana, desde que satisfaça uma certa condição integral. Prosseguindo, mostraremos que as bolas geodésicas de metricas quasi-Einstein est áveis não compactas tem crescimento no mí nimo linear. Finalmente, usaremos métrica quasi-Einstein, para provarmos um teorema de trivialidade para uma certa classe de produto warped Einstein, sob uma hipótese que envolve a função warped e as constantes de Einstein do produto warped e da fibra.

CARLOS, Elaine Sampaio de Sousa. A geometria dos sólitons de Ricci compactos. 2013. 44 f. Dissertação (Mestrado em Matemática) - Centro de Ciências, Universidade Federal do Ceará, Programa de Pós-Graduação em Matemática, Fortaleza, 2013 / Submitted by Erivan Almeida (eneiro@bol.com.br) on 2014-02-06T12:21:20Z No. of bitstreams: 1 2013_dis_esscarlos.pdf: 420299 bytes, checksum: 8818a543bee71f541828a3b7c3035f0c (MD5) / Approved for entry into archive by Rocilda Sales(rocilda@ufc.br) on 2014-02-06T13:33:06Z (GMT) No. of bitstreams: 1 2013_dis_esscarlos.pdf: 420299 bytes, checksum: 8818a543bee71f541828a3b7c3035f0c (MD5) / Made available in DSpace on 2014-02-06T13:33:06Z (GMT). No. of bitstreams: 1 2013_dis_esscarlos.pdf: 420299 bytes, checksum: 8818a543bee71f541828a3b7c3035f0c (MD5) Previous issue date: 2013 / The aim of this work is to study the geometry of the compact Ricci soliton, which correspond to self-similar solution of the Ricci flow. These manifolds are natural generalization to Einstein metrics. Here we shall prove that every compact Ricci soliton has positive scalar curvature. Moreover, we show that its fundamental group is finite. Finally, we prove that every compact Ricci soliton must be gradient. / O objetivo deste trabalho é estudar a geometria dos sólitons de Ricci compactos, os quais correspondem as soluções auto-similires do fluxo de Ricci. Além disso, essas variedades podem ser vistas como uma generalização das métricas de Einstein. Neste trabalho, mostraremos que todo sóliton de Ricci compacto tem curvatura escalar positiva. Além disso, mostraremos que o seu grupo fundamental é sempre finito. Em particular, apresentaremos uma prova feita por Perelman [19] que todo sóliton de Ricci compacto é do tipo gradiente.

CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / Neste trabalho estudamos hipersuperfÃcies mÃnimas completas e com curvatura de Gauss-Kronecker constante em uma forma espacial Q4(c). Provamos que o Ãnfimo do valor absoluto da curvatura de Gauss-Kronecker de uma hipersuperfÃcie mÃnima completa em Q4(c); c ≤ 0; na qual a curvatura de Ricci à limitado inferiormente, à igual a zero. AlÃm disso, estudamos hipersuperfÃcies mÃnimas conexas M3 em uma forma espacial Q4(c) com curvatura de Gauss-Kronecker K constante. Para o caso c ≤ 0, provamos, por um argumento local, que se K à constante, entÃo K deve ser igual a zero. TambÃm apresentamos uma classificaÃÃo de hipersuperfÃcies completas mÃnimas em Q4 com K constante. Exemplos de hipersuperfÃcies mÃnimas que nÃo sÃo totalmente geodÃsicas no espaÃo Euclidiano e no espaÃo hiperbÃlico com curvatura de Gauss-Kronecker nula sÃo apresentados. / In this work we study complete minimal hypersurfaces with constant Gauss-Kronecker curvature in a space form Q4(c). We prove that the infimum of the absolute value of the Gauss-Kronecker curvature of a complete minimal hypersurface in Q4(c); c ≤ 0; whose Ricci curvature is bounded from below,is equal to zero. Futher, we study the connected minimal hypersurfaces M3 of a space form Q4(c) with constant Gauss-Kronecker curvature K. For the case c ≤ 0, we prove, by a local argument, that if K is constant, then K must be equal to zero. We also present a classification of complete minimal hypersurface of Q4 with K constant. Examples of complete minimal hypersurfaces which are not totally geodesic in the Euclidean space R4 and the hiperbolic space H4(c) with vanishing Gauss-Kronecker curvature are also presented.

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.

In the first part of this thesis, we proved a pseudo-locality theorem for a coupled Ricci flow, extending Perelman’s work on Ricci flow to the Ricci flow coupled with heat equation. By use of the reduced distance and the pseudo-locality theorem, we showed that the parabolic rescaling of a Type I coupled Ricci flow with respect to a Type I singular point converges to a non-trivial Ricci soliton. In the second part of the thesis, we prove the existence of infinitely many solutions to the Hull- Strominger system on generalized Calabi-Gray manifolds, more specifically compact non-K \"ahler Calabi-Yau 3-folds with infinitely many distinct topological types and sets of Hodge numbers. We also studied the behavior of the anomaly flow on the generalized Calabi-Gray manifolds, and reduced it to a scalar flow on a Riemann surface. We obtained the long-time existence and convergence after rescaling in the case when the curvature of initial metric is small.