张晓林. / 呈交日期: 2002年7月. / 论文(哲学博士)--香港中文大学, 2003. / 参考文献 (p. 164-177). / 中英文前言. / Cheng jiao ri qi: 2002 nian 7 yue. / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. Ann Arbor, MI : ProQuest Information and Learning Company, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Mode of access: World Wide Web. / Zhang Xiaolin. / Zhong Ying wen qian yan. / Lun wen (zhe xue bo shi)--Xianggang Zhong wen da xue, 2003. / Can kao wen xian (p. 164-177).

CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / Esta tese teve dois objetos de estudo: propriedades estocÃsticas em uma variedade Riemanniana, a saber, Completude EstocÃstica, Parabolicidade e propriedade Feller, e a geometria do tensor de Bakry-Emery. Na primeira parte da tese estudamos tais propriedades estocÃsticas no contexto de submersÃes Riemannianas e imersÃes isomÃtricas, tendo como ponto de partida o trabalho de Pigola e Setti [28] sobre a propriedade Feller. No nosso primeiro resultado, provamos que se uma imersÃo isomÃtrica em uma variedade Cartan-Hadamard possui vetor curvatura mÃdia com norma limitada entÃo a imersÃo à Feller. Um anÃlogo desse resultado jà era conhecido para o caso de completude estocÃstica [30]. Em seguida estabelecemos condiÃÃes necessÃrias e suficientes para que uma submersÃo seja estocasticamente completa (respec. parabÃlica), a saber, se uma submersÃo Riemanniana tem fibra mÃnima e o espaÃo total à estocasticamente completo (respec. parabÃlico) entÃo a base à estocasticamente completa (respec. parabÃlica). Reciprocamente, se a submersÃo Riemanniana tem fibra mÃnima e compacta e a base à estocasticamente completa (respec. parabÃlica) entÃo o espaÃo total à estocasticamente completo (respec. parabÃlico). Finalmente provamos que uma submersÃo Riemanniana tem fibra mÃnima e compacta entÃo o espaÃo total Âe Feller, se, e somente se, a base à Feller. Na segunda parte desta tese estudamos o tensor de Bakry-Emery Ricci, Ricf, que à uma extensÃo, no caso de variedades ponderadas, do tensor de Ricci. Estudamos a seguinte situaÃÃo: Ricci ≥ -cG, onde c à uma constante positiva e G ≥ O à uma funÃÃo suave. Esta limitaÃÃo nos permitiu obter algumas consequencias geomÃtricas e topolÃgicas, que passamos a descrever. Seja Mf uma variedade Riemanniana ponderada e po Є Mf fixado. Nosso primeiro resultado à uma estimativa superior, fora da bola geodÃsica de raio ro, para o Laplaciano ponderado da funÃÃo distÃncia r ao ponto po, mf, em termos da integral da funÃÃo G. A primeira consequÃncia dessa estimativa à uma estimativa para o volume ponderado Volf (B(R)) de uma bola geodÃsica de raio R em termos da integral da funÃÃo G. A estimativa de mf, juntamente com a hipÃtese de Æ ser radial e Әr Æ ≥ -a,a ≥ 0 (ou | Æ|≤ k) tambÃm nos permite demonstrar um teorema de comparaÃÃo entre mf e maG, Laplaciano da funÃÃo distÃncia no modelo de curvatura aG, bem como um teorema de comparaÃÃo entre o volume ponderado de uma bola geodÃsica de raio R em Mf, VolÆ(B(R)), e o volume da bola geodÃsica de raio R no modelo MaG, de curvatura aG. Utilizando uma versÃo ponderada da fÃrmula de Bochner provamos que, se Ricci ≥ Gâ entÃo Mf satisfaz o princÃpio do mÃximo de Omori-Yau, onde G à funÃÃo suave, positiva, nÃo decrescente e tal que G-1 nÃo à integrÃvel. Em particular concluÃmos que Mf à estocasticamente completa. O prÃximo resultado que obtivemos estende, para o tensor Ricf, um teorema de Myers devido a Ambrose [1]. Para tanto, uma hipÃtese sobre a funÃÃo Æ foi necessÃria. Como aplicaÃÃo, estendemos um resultado de compacidade de Ricci solitons de Fernando-Lopes e Garcia-Rio [15]. Em 1976, Yau [36] provou uma estimativa para o gradiente de uma funÃÃo u, positiva, harmÃnica em B(2R), no caso de M ser completa e Ricf ≥ -k, k ≥ 0. Tal estimativa depende apenas de R e k e foi estendida, no caso ponderado, para funÃÃes f harmÃnicas positivas, supondo Ricf ≥ -k e Ric ≥ -H, k, H ≥ 0. Bringhton [9] obteve estimativas para o gradiente de uma funÃÃo *-harmÃnica positiva utilizando somente a hipÃtese Ricf ≥ -k. As estimativas que obtivemos estendem as estimativas citas acima e, no caso em que Æ=G=0 resultam na estimativa original de Yau. Finalmente, provamos um teorema de comparaÃÃo entre o primeiro autovalor de Dirichlet da bola geodÃsica de raio R em Mf e o primeiro autovalor de Dirichlet da bola geodÃsica de raio MG. Tal resultado estende, para o caso ponderado, um resultado de Bessa e Montenegro [4]. / In this thesis we studied two objects(?): properties in Riemannian manifolds, more precisely stochastic completeness, parabolicity and the Feller property and geometric properties of Bakry Emery Ricci tensor. First, we studied such stochastic properties on Riemannian and isometric immersions. The initial motivation was the work of Pigola and Setti [30] about the Feller property. In our first result, we proved that if a isometric immersion on a Cartan-Hadamard manifold has bounded mean curvature vector then the immersion is Feller. An analogous result was know for stochastic completeness. After we stabilish necessary and sufficient conditions to a Riemannian submersion be stochastically complete (parabolic). More precisely if a Riemannian submersion has minimal fiber and the total space is stochastically complete (parabolic ) then the basis is also stochastically complete ( parabolic ). Conversely, if the Riemannian submersion has compact minimal fiber and the basis is stochastically complete ( parabolic, Feller ) then the total space also is. We also proved that if a Riemannian submersion has compact minimal fiber then the total space is Feller if, and only if the the basis is Feller. In the second part we studied the Barkry Emery Ricci tensor Ricf, wich is a natural extension of the Ricci tensor in the context of weighted manifolds. We studied the following: suppose that Ricf has a lower bound âcG where G is a smooth nonnegative function and c a positive constant. Such lower bound allow us to obtain some geometric and topological consequences as we describe below. Consider Mf a weighted Riemannian manifold. The first consequence is an upper estimate, outside a geodesic ball of radius r0, for the weighted Laplacian of the Riemannian distance in terms of the function G. Let Mf be a weighted Riemannian manifold and po Є Mf fixed. Our first result is an upper bound, outside of a geodesic ball of radius R centered in po, for the weighted Laplacian os the Riemannian distance function from po in terms od the function G. The first consequence of this estimate is an estimate for the weighted volume Volf (B(R)) of a geodesic ball with radius R in terms of the integral of G. This estimate together the assumption of f be radial and Ә f ≥ - a, a≥ 0 (or | f | ≤k ) allow us to prove a comparison theorem for mf e mag, the Laplacian of distance function of the Riemannian model fo curvature aG, as such as a comparison theoremfor the weighted volume of a geodesic ball with radius R on the Riemannian model MaG, with curvature aG. Using a weighted version of the Bochner formula we proved that Ricf ≥ Gâ then Mf satisfies the Omori-Yau Maximum Principle, where G is a positive, nondecreasing smooth function, such that G-1 does not belong to L1(Mf). In particular we conclude that Mf is stochastically complete. The next result we proved extends, for the tensor Ricf, a type Myers theorem due to Ambrose [1]. For this an additional assumption on f was required. As an aplication of this result we extended a result about compacity of Ricci solitons due to Fernandez-Lopez e GarcÃa-Rio [15]. In 1976, Yau [36] proved an estimate for the gradient of a positive harmonic funcion u, defined on B(2R), when M is complete and Ric ≥ -k, k≥ 0. Such estimate depends only on R and k and was extended, to the weighted, to the case, to f-harmonic positive functions, when Ricf ≥ - k and Ric ≥ - H, k, H ≥ 0. Brighton [9] obtained estimates for the gradient of a positive f-harmonic function assuming only Ricf ≥ -k. We obtained estimates for the case Ricf ≥ -G where G is a smooth nonnegative function and when f= G = 0 we recover the original estimate of Yau. Finally we proved a comparison theorem between the first eigenvalue of the geodesic ball of radius r on Mf and the first eigenvalue of the geodesic ball of radius r of the model MG. Such result extends, to the weighted case, a result due to Bessa e Montenegro [4].

This thesis deals with Einstein metrics and the Ricci flow on compact mani- folds. We study the second variation of the Einstein-Hilbert functional on Ein- stein metrics. In the first part of the work, we find curvature conditions which ensure the stability of Einstein manifolds with respect to the Einstein-Hilbert functional, i.e. that the second variation of the Einstein-Hilbert functional at the metric is nonpositive in the direction of transverse-traceless tensors. The second part of the work is devoted to the study of the Ricci flow and how its behaviour close to Einstein metrics is influenced by the variational be- haviour of the Einstein-Hilbert functional. We find conditions which imply that Einstein metrics are dynamically stable or unstable with respect to the Ricci flow and we express these conditions in terms of stability properties of the metric with respect to the Einstein-Hilbert functional and properties of the Laplacian spectrum. / Die vorliegende Arbeit beschäftigt sich mit Einsteinmetriken und Ricci-Fluss auf kompakten Mannigfaltigkeiten. Wir studieren die zweite Variation des Einstein- Hilbert Funktionals auf Einsteinmetriken. Im ersten Teil der Arbeit finden wir Krümmungsbedingungen, die die Stabilität von Einsteinmannigfaltigkeiten bezüglich des Einstein-Hilbert Funktionals sicherstellen, d.h. die zweite Varia- tion des Einstein-Hilbert Funktionals ist nichtpositiv in Richtung transversaler spurfreier Tensoren. Der zweite Teil der Arbeit widmet sich dem Studium des Ricci-Flusses und wie dessen Verhalten in der Nähe von Einsteinmetriken durch das Variationsver- halten des Einstein-Hilbert Funktionals beeinflusst wird. Wir finden Bedinun- gen, die dynamische Stabilität oder Instabilität von Einsteinmetriken bezüglich des Ricci-Flusses implizieren und wir drücken diese Bedingungen in Termen der Stabilität der Metrik bezüglich des Einstein-Hilbert Funktionals und Eigen- schaften des Spektrums des Laplaceoperators aus.

This thesis is twofold. In the first part, a proof of the interpolation inequality along geodesics in p-Wasserstein spaces is given and a new curvature condition on abstract metric measure spaces is defined. In the second part of the thesis a proof of the identification of the q-heat equation with the gradient flow of the Renyi (3-p)-Renyi entropy functional in the p-Wasserstein space is given. For that, a further study of the q-heat flow is presented including a condition for its mass preservation.

Dans cette thèse, nous étudions principalement les applications semi-conformes et leur influence sur la résolution de certaines équations géométriques importantes comme celle d’un soliton de Ricci et celle d’une application biharmonique. Dans la première partie, nous appliquons un ansatz qui permet de construire des applications semi-conformes à partir d’une équation différentielle en une fonction de deux variables. Nous caractérisons les solutions réelles-analytiques. Parmi les solutions explicites obtenues, nous trouvons le premier exemple d’une application semi-conforme non-harmonique définie entièrement sur R3 à valeurs dans le plan complexe. Dans la deuxième partie, nous étudions les solitons de Ricci. Nous nous intéressons aux solitons de dimension 3, où ils peuvent être décrits, au moins localement, en terme d’une application semi-conforme. Nous développons une nouvelle méthode de construction de ces solitons à partir des transformations biconformes, particulièrement adaptées à l’étude de l’unicité de la structure. Finalement, nous introduisons une nouvelle notion de morphisme harmonique généralisé qui, comme son nom l’indique, contient les morphismes harmoniques comme un cas particulier. Cette classe d’applications a une importance dans la théorie d’applications biharmoniques. Les morphismes harmoniques généralisés ont une caractérisation nette qui permet de donner plusieurs exemples et méthodes de construction d’applications biharmoniques non-harmonique. / In this work, we primarily study semiconformal mappings and their influence in the resolution of important geometric equations, such as those for a Ricci soliton and those for a biharmonic maps. In the first part of this thesis, we exploit an ansatz for the construction of semi-conformal mappings from a differential equation in a function of two variables. We characterize real-analytic solutions.Among the resulting explicit solutions, we find the first known example of an entire semi-conformal mapping into the plane which is not harmonic. In the second part, we study Ricci solitons.We are particularly interested in 3-dimensional Ricci solitons, as they can be described at least locally, in terms of a semi-conformal map. We develop a construction method of solitons from biconformal deformations, particularly adapted to the study of the structure unicity. Finally, we introduce a new notion of generalized harmonic morphism, which, as the name suggests, contain the harmonic morphisms as a special case. These mappings have an elegant characterization which enables the construction of explicit examples, as well as impacting on the theory of biharmonic mappings.

Submitted by Jaqueline Silva (jtas29@gmail.com) on 2014-10-23T20:04:48Z No. of bitstreams: 2 Dissertação - Hiuri Fellipe Santos dos Reis - 2013.pdf: 1601406 bytes, checksum: f2663891a9c0968329f2f913ada41d9e (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Approved for entry into archive by Jaqueline Silva (jtas29@gmail.com) on 2014-10-23T20:05:03Z (GMT) No. of bitstreams: 2 Dissertação - Hiuri Fellipe Santos dos Reis - 2013.pdf: 1601406 bytes, checksum: f2663891a9c0968329f2f913ada41d9e (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Made available in DSpace on 2014-10-23T20:05:03Z (GMT). No. of bitstreams: 2 Dissertação - Hiuri Fellipe Santos dos Reis - 2013.pdf: 1601406 bytes, checksum: f2663891a9c0968329f2f913ada41d9e (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Previous issue date: 2013-03-22 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work we present a study on locally conformally flat gradient steady Ricci solitons which is based on a Huai Dong-Cao and Qing Chen’s article, where they was classified the n-dimensional (n ≥ 3) complete noncompact locally conformally flat gradient steady Ricci solitons. In particular, we prove that a complete noncompact non-flat locally conformally flat gradient steady Ricci soliton is, up to scaling, the Bryant soliton. / Neste trabalho apresentamos um estudo dos sólitons de Ricci gradiente steady localmente conformemente flat, baseado no trabalho de Huai-Dong Cao e Qiang Chen, onde são classificados os sólitons de Ricci gradiente steady n-dimensionais (n ≥ 3), completos, não-compactos e localmente conformemente flat. Em particular provamos que um sóliton de Ricci gradiente steady completo, não-compacto, não-flat e localmente conformemente flat é, a menos de homotetia, o sóliton de Bryant.

Submitted by Luciana Ferreira (lucgeral@gmail.com) on 2015-11-30T07:33:27Z No. of bitstreams: 2 Tese - Márcio Lemes de Sousa - 2015.pdf: 2626758 bytes, checksum: 1e9e1b9d216bad33d6b5919afa54a4e4 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2015-11-30T07:35:41Z (GMT) No. of bitstreams: 2 Tese - Márcio Lemes de Sousa - 2015.pdf: 2626758 bytes, checksum: 1e9e1b9d216bad33d6b5919afa54a4e4 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Made available in DSpace on 2015-11-30T07:35:41Z (GMT). No. of bitstreams: 2 Tese - Márcio Lemes de Sousa - 2015.pdf: 2626758 bytes, checksum: 1e9e1b9d216bad33d6b5919afa54a4e4 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Previous issue date: 2015-07-03 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this thesis, primarily, we studied warped products semi-Riemannian Einstein manifolds. We considered the case in that the base is conformal to an n-dimensional pseudo- Euclidean space and invariant under the action of an (n 􀀀 1)-dimensional translation group. We constructed new examples of Einstein warped products with zero Ricci curvature when the fiber is Ricci-flat. In particular, we obtain explicit solutions, in the case vacuum, for Einstein field equation. Furthermore, we consider M = B f F warped product gradient Ricci solitons. We proved that the potential function depends only on the base and the fiber F is necessarily Einstein manifold. We provide all such solutions in the case of steady gradient Ricci solitons when the base is conformal to an n-dimensional pseudo-Euclidean space, invariant under the action of an (n􀀀1)-dimensional translation group, and the fiber F is Ricci-flat. / Nesta tese, primeiramente, estudamos variedades produto torcido semi-Riemannianas de Einstein, considerando-se o caso em que a base é conforme ao espaço pseudo- Euclidiano n -dimensional e invariante sob a ação de um grupo de translações (n􀀀1)-dimensional. Construímos novos exemplos de métricas produto torcido Einstein com curvatura de Ricci zero quando a fibra é Ricci -flat. Em particular, obtemos soluções explícitas, no caso de vácuo, para a equação de campo de Einstein. Em seguida, provamos que quando a variedade M = B f F é um Ricci soliton gradiente a função potencial depende apenas da base e a fibra F é necessariamente uma variedade de Einstein. Fornecemos todas as soluções, no caso de Ricci soliton gradiente steady, quando a base é conforme ao espaço pseudo- Euclidiano n -dimensional, invariante sob a ação de um grupo translações (n􀀀1) - dimensional, e a fibra F é Ricci -flat.

Submitted by Marlene Santos (marlene.bc.ufg@gmail.com) on 2016-06-15T19:51:42Z No. of bitstreams: 2 Dissertação - Elismar Dias Batista - 2016.pdf: 1518873 bytes, checksum: 8375db389a2056c5849ee168f5efa5ce (MD5) license_rdf: 19874 bytes, checksum: 38cb62ef53e6f513db2fb7e337df6485 (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2016-06-28T12:21:16Z (GMT) No. of bitstreams: 2 Dissertação - Elismar Dias Batista - 2016.pdf: 1518873 bytes, checksum: 8375db389a2056c5849ee168f5efa5ce (MD5) license_rdf: 19874 bytes, checksum: 38cb62ef53e6f513db2fb7e337df6485 (MD5) / Made available in DSpace on 2016-06-28T12:21:16Z (GMT). No. of bitstreams: 2 Dissertação - Elismar Dias Batista - 2016.pdf: 1518873 bytes, checksum: 8375db389a2056c5849ee168f5efa5ce (MD5) license_rdf: 19874 bytes, checksum: 38cb62ef53e6f513db2fb7e337df6485 (MD5) Previous issue date: 2016-03-31 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / This work is based on the articles [26] and [27], where we studied Einstein manifolds and gradient Ricci soliton with twisted product structure. As a result, we prove the following: if M is an Einstein warped product space with nonpositive scalar curvature and compact base, then M is a Riemannian product space. Besides, we show that the Riemannian product Rp×F is a gradient Ricci soliton if and only if F is Ricci soliton gradient. Then, we show that the warped product R×f B is gradient Ricci solitons with f ′′ 6= 0, therefore F is Einstein. By using these results, we build nontrivial examples of gradient Ricci soliton where the fiber is either an Einstein manifold or a nontrivial gradient Ricci soliton. / Este trabalho está baseado nos artigos [26] e [27], onde estudamos Variedades de Einstein e gradiente Ricci solitons com estrutura de produto torcido. Provamos que: se M é um produto torcido Einstein com curvatura escalar não positiva e base compacta, então a função torção é constante, ou seja, o produto torcido é Riemanniano. Mostramos ainda que o produto Riemanniano Rp ×F é um gradiente Ricci soliton se e somente se F for gradiente Ricci soliton. Em seguida, mostramos que se o produto torcido R×f F for gradiente Ricci soliton com f ′′(t) 6= 0, então F é Einstein. Usando estes resultados construímos exemplos de gradiente Ricci soliton não trivial com a fibra sendo Einstein ou gradiente Ricci soliton não trivial. Finalmente consideramos o produto torcido Lorentziano sendo gradiente Ricci soliton e obtivemos critérios análogos ao Riemanniano para que F seja Einstein ou gradiente Ricci soliton.

Submitted by Jaqueline Silva (jtas29@gmail.com) on 2016-11-09T17:13:45Z No. of bitstreams: 2 Dissertação - Ilton Ferreira de Menezes - 2016.pdf: 1261743 bytes, checksum: d8e7ce96b09f78e6c7a8c4d534a9d401 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Jaqueline Silva (jtas29@gmail.com) on 2016-11-09T17:13:57Z (GMT) No. of bitstreams: 2 Dissertação - Ilton Ferreira de Menezes - 2016.pdf: 1261743 bytes, checksum: d8e7ce96b09f78e6c7a8c4d534a9d401 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2016-11-09T17:13:57Z (GMT). No. of bitstreams: 2 Dissertação - Ilton Ferreira de Menezes - 2016.pdf: 1261743 bytes, checksum: d8e7ce96b09f78e6c7a8c4d534a9d401 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2016-10-14 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / This work is based on [10] and aims to classify quasi-Einstein manifolds that are locally conformally flat. We prove that every complete, locally conformally flat, quasi-Einstein manifold, with dimension n ≥ 3, is either globally conformally equivalent to spaceform or locally the warped product, R×Ffn−1, in which the fiber has constant curvature. / Este trabalho está baseado em [10] e tem por objetivo classificar variedades quasi- Einstein que são localmente conformemente planas. Provamos que toda variedade quasi- Einstein localmente conformente plana, completa e de dimensão n ≥ 3 é globalmente conformemente equivalente a um dos espaços modelos ou é localmente o produto torcido R×Ffn−1 onde a fibra tem curvatura constante.

Submitted by JÚLIO HEBER SILVA (julioheber@yahoo.com.br) on 2017-05-04T20:17:22Z No. of bitstreams: 2 Dissertação - Maria Francisca de Sousa Gomes - 2017.pdf: 1138083 bytes, checksum: ec11ffa7d803dc5e840f5b216f1aaba3 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2017-05-05T13:03:10Z (GMT) No. of bitstreams: 2 Dissertação - Maria Francisca de Sousa Gomes - 2017.pdf: 1138083 bytes, checksum: ec11ffa7d803dc5e840f5b216f1aaba3 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2017-05-05T13:03:10Z (GMT). No. of bitstreams: 2 Dissertação - Maria Francisca de Sousa Gomes - 2017.pdf: 1138083 bytes, checksum: ec11ffa7d803dc5e840f5b216f1aaba3 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2017-04-20 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / This work is based on [1] and aims to show a result of rigidity for gradient almost Ricci soliton. We will prove that an almost Ricci soliton gradient with nonnegative scalar curvature, where ∇ f is a non-trivial conformal field, is either a Euclidean space R n or the sphere S n . Moreover, we have that, in the Spherical case, the potential function is given by first eigenfunction of the Laplacian. Finally, we will find necessary and sufficient conditions for that a compact locally conformally flat gradient almost Ricci soliton is isometric the sphere Sn. / Este trabalho está baseado em [1] e tem por objetivo apresentar um resultado de rigidez para gradiente quase Ricci soliton. Provaremos que um gradiente quase Ricci soliton com curvatura escalar não-negativa, em que ∇ f é um campo conforme não-trivial, é ou o espaço Euclidiano R n ou a Esfera S n . Além disso, temos que no caso Esférico, a função potencial é dada pela primeira auto função do Laplaciano. Por fim, encontraremos condições necessárias e suficientes para que um gradiente quase Ricci soliton compacto localmente conformemente flat seja isométrico a esfera Sn.