SANTOS, A. S. L. Problema de Yamabe modificado em variedades compactas de dimensão quatro e métricas críticas do funcional curvatura escalar. 2017. 58 f. Tese (Doutorado em Matemática) – Centro de Ciências, Universidade Federal do Ceará, Fortaleza, 2017. / Submitted by Andrea Dantas (pgmat@mat.ufc.br) on 2017-05-25T19:34:47Z No. of bitstreams: 1 2017_tese_aslsantos.pdf: 535461 bytes, checksum: 8c3ddbdd33d74c4eb7b265354b3bafb3 (MD5) / Rejected by Rocilda Sales (rocilda@ufc.br), reason: Boa tarde, Eu revisei a Tese de ALEX SANDRO LOPES SANTOS, e encontrei um pequeno erro na capa, ele colocou os seguintes elementos: UNIVERSIDADE FEDERAL DO CEARÁ CENTRO DE CIÊNCIAS PROGRAMA DE PÓS-GRADUAÇÃO EM MATEMÁTICA DOUTORADO EM MATEMÁTICA Mas deve ser alterado para: UNIVERSIDADE FEDERAL DO CEARÁ CENTRO DE CIÊNCIAS DEPARTAMENTO DE MATEMÁTICA PROGRAMA DE PÓS-GRADUAÇÃO EM MATEMÁTICA Com os demais elementos da Tese, não há nenhum problema de formatação. Atenciosamente, on 2017-05-26T15:06:03Z (GMT) / Submitted by Andrea Dantas (pgmat@mat.ufc.br) on 2017-05-29T13:47:44Z No. of bitstreams: 1 2017_tese_aslsantos.pdf: 536279 bytes, checksum: f37ece7d8035a2d9c788c45d2e7807ae (MD5) / Approved for entry into archive by Rocilda Sales (rocilda@ufc.br) on 2017-05-29T14:08:17Z (GMT) No. of bitstreams: 1 2017_tese_aslsantos.pdf: 536279 bytes, checksum: f37ece7d8035a2d9c788c45d2e7807ae (MD5) / Made available in DSpace on 2017-05-29T14:08:17Z (GMT). No. of bitstreams: 1 2017_tese_aslsantos.pdf: 536279 bytes, checksum: f37ece7d8035a2d9c788c45d2e7807ae (MD5) Previous issue date: 2017-05-19 / In the fisrt part of this work we investigate the modified Yamabe problem on four-dimensional manifolds whose the modifiers invariants depending on the eigenvalues of the Weyl curvature tensor and they are described in terms of maximum and minimum of the biorthogonal (sectional) curvature. We provide some geometrical and topological properties on four-dimensional manifolds in terms of these invariants. In the second part we investigate the critical points of the total scalar curvature functional restricted to space of metrics with constant scalar curvature of unitary volume, for simplicity CPE metrics. It was conjectured in the 1980’s that every CPE metric must be Einstein. We prove that such a conjecture is true under a second-order vanishing condition on the Weyl tensor. / Na primeira parte deste trabalho investigamos o problema de Yamabe modificado em variedades de dimensão quatro cujos invariantes modificadores dependem dos autovalores do tensor de Weyl e são descritos em termos do máximo e mínimo da curvatura biortogonal (seccional). Fornecemos algumas propriedades geométricas e topológicas para tais variedades em termos destes invariantes. Na segunda parte investigamos os pontos críticos do funcional curvatura escalar total restrito ao espaço de métricas com curvatura escalar constante e volume unitário, abreviadamente chamamos de métricas CPE. Conjecturou-se na década de 1980 que toda métrica CPE deve ser Einstein. Provamos que tal conjectura é verdadeira sob uma condição de nulidade sobre o divergente de segunda ordem do tensor de Weyl.

Nanotecnologia é o conjunto de conhecimentos acumulados pelo homem que permite controlar a produção de estruturas com uma ou mais dimensões nanométricas. Desde seus primórdios na década de 70, a nanotecnologia tem estado em constante ascensão, encontrando uma diversidade enorme de aplicações, como por exemplo em medicina e na indústria optoeletrônica. Por consequência, a demanda por equipamentos tanto de preparo como de caracterização/controle tem crescido exponencialmente. O uso da radiação X no estudo de dispositivos nanoestruturados tem sido, em grande parte, possível gra- ças as fontes síncrotrons com feixes intensos. Mas a disponibilidade desses laboratórios de alta tecnologia está aquém da crescente demanda das pesquisas em nanotecnologia, as quais precisam de técnicas de análise estrutural rápidas e de fácil acesso para otimização e controle da produção de dispositivos nanoestruturados. Com foco nessa falta por técnicas de análise estrutural, esta dissertação tem como objetivo avaliar quais parâmetros básicos de nanodispositivos, com substratos monocristalinos, podem ser investigados por meio de técnicas de difração de raios X utilizando fontes compactas de radiação, bem como avaliar as limitações instrumentais. / X-ray radiation has provided a powerful tool for analyzing the structure of materials at atomic scale. While many are fascinated with the perspectives oered by advanced X-ray sources, the practical aspects of these perspectives in the actual and future scenery of nanotechnology needs to be discussed. Nanotechnology, i.e. the capacity of controlling matter at atomic-molecular scales and manufacturing structures with dimensions of a few tens of nanometers, has provided a constant challenge for structural analysis via X-ray techniques. The great diversity of materials and methods derived from nanotechnology is generating a huge demand for time of analysis, much beyond of that can be supplied by synchrotron facilities worldwide. In optimizing nanostructured materials and devices processing methods, fast and easy-access techniques to control and characterization are required. Microscopy and spectroscopy techniques are very important in this scenery, but they have intrinsic limitations that have justied the search for high-resolution techniques of structural analysis, such as those obtained by diraction of X-rays. The use of Xradiation in the study of nanostructured device has been possible by synchrotron sources due to the high intense beams. But the availability of these high tech laboratories falls short of the growing demand for nanotechnology research. In this context, this dissertation intend evaluate which basic parameters of nanodevices with single crystal substrates can be investigated by techniques of X-ray diraction using compact radiation sources as well how to evaluate the instrumental limitations .

Made available in DSpace on 2016-04-15T23:44:02Z (GMT). No. of bitstreams: 2 Claudia Pauperio Titton2.pdf: 21379590 bytes, checksum: 008075a69606e0c7342c3b4b6b4e1af2 (MD5) Claudia Pauperio Titton1.pdf: 30008290 bytes, checksum: df9b9d0ed74f0113d1d0b409f38176b4 (MD5) Previous issue date: 2012-06-01 / Fundo Mackenzie de Pesquisa / Among the many urban consequences caused by the process of global economic restructuring, there is the emergence of large idle areas in cities and urban sectors whose growth had once been supported by the manufacturing industry. These are called areas of demand for industrial conversion or positive restructuring. The territory of the Fourth District of Porto Alegre, the object of the research study, is inserted in this context and problem set, offering many opportunities for new urban projects as a tool for territorial regeneration of degraded or transforming land. The urbanization process, which is currently characterized by the ability to put parts of the city in motion, has the challenge of dealing with existing city, without denying it. Similarly, society enters the twenty-first century with a strong concern for environmental preservation and recycling of resources, demanding the transformation and recovery of environments, such as industrial ones, through sustainable development. Thus, the paper focuses on aspects of the factors present in the origin of these places, their impacts on the land, the possibilities within the productive restructuring and urban regeneration of traditionally industrial areas, their new formats in the contemporary city - amongst them, the urban clusters, new fields of architecture and urban design facing society and the contemporary cities, sustainable development and condensing the city through the (re) densification of such territories, and finally focuses on the analysis and diagnosis of the context and potential of the IV District. The paper seeks to illustrate the theoretical discussion with relevant project experience. / Entre as inúmeras conseqüências urbanas provocadas pelo processo de reestruturação econômica global, destaca-se o surgimento de grandes áreas ociosas nas cidades e setores urbanos cujo crescimento havia se amparado na indústria de transformação. São as chamadas áreas de demanda por reconversão industrial ou reestruturação produtiva. É nesse contexto que está inserido o território do IV Distrito de Porto Alegre, objeto de estudo da pesquisa, e, nessa problemática, encontram-se muitas das oportunidades oferecidas aos novos projetos urbanos, enquanto instrumentos de regeneração territorial de áreas produtivas deterioradas ou em transformação - o processo urbanístico, que atualmente caracteriza-se pela capacidade de colocar trechos da cidade em movimento, tem o desafio de enfrentar a cidade existente, sem negá-la. Paralelamente, a sociedade adentra o século XXI com a forte preocupação da preservação ambiental e da reciclagem dos recursos disponíveis, fazendo com que a transformação e a recuperação de ambientes, como os industriais, coloquem-se dentro dessa nova demanda através do desenvolvimento sustentável. Sendo assim, o trabalho aborda aspectos acerca dos fatores presentes na origem desses lugares, de seus impactos no território, das possibilidades existentes na reestruturação produtiva e na regeneração urbana de zonas tradicionalmente industriais, de seus novos formatos na cidade contemporânea - entre eles, os clusters urbanos, dos novos campos de ação da arquitetura e do projeto urbano frente à sociedade e às cidades contemporâneas, do desenvolvimento sustentado e da promoção da cidade compacta através da (re)densificação desses territórios, e por fim, concentra-se na análise e no diagnóstico do contexto e das potencialidades do IV Distrito, sempre buscando ilustrar a discussão teórica com experiências projetuais relevantes que colaborem com o esclarecimento das questões encontradas na região em estudo e que inspirem alternativas para sua reversão.

Made available in DSpace on 2016-04-16T00:14:36Z (GMT). No. of bitstreams: 2 Gustavo Luiz Mestriner1.pdf: 2243793 bytes, checksum: a1c06f5eaa7eff79bc276f72d6e0e06e (MD5) Gustavo Luiz Mestriner2.pdf: 2750701 bytes, checksum: a5abda0e155ae8b1a195f0e2701c63f1 (MD5) Previous issue date: 2008-08-12 / Cities all over the world, go through a process of urban changing never seen before. Data from great developing population in cities, added with the urban exodus from rural areas, are the causes of swelling, and later, an explosive urban sprawlling caused in cities. In this process, the land appreciation in urban centers and tax incentives and the cheaper labor in neighboring towns, made that big industries change from urban centers, leaving the obsolete large areas. On the other hand, the cities, originally created for the celebration of life, and to celebrate what we have in common, today are draining energy resources, and the most polluting source of the planet. The new urban projects should consider this complex reality, then look for some contemporaries principles, such as sustainable urban development, productive restructuring, the environments of research and technological innovation and biotechnology. To do so, development agiencies have been created to conciliate the public and private interests. We explore the concept of compact city, in the light of contemporary urban projects in wastelands in central areas. To context, we selected some case studies, building an analytical inventory and always relating them to our context and reality. / Cidades pelo mundo inteiro passam por um processo de mutação urbana jamais visto. Os dados do desenvolvimento substancial demográfico nas cidades, somados aos êxodos urbanos oriundos das zonas rurais são as causas do inchaço, e posterior espraiamento urbano explosivo causado nas cidades. Neste processo, a valorização da terra nos centros urbanos e o incentivo fiscal e de mão de obra mais barata nas cidades vizinhas, fez com que as grandes indústrias se deslocassem dos centros urbanos, deixando grandes áreas obsoletas. Por outro lado, as cidades, inicialmente criadas para a celebração da vida, e para celebrar o que temos em comum, hoje são poços drenantes de recursos energéticos, e direta e indiretamente, a maior fonte poluidora do planeta. Desta forma, os novos projetos urbanos devem levar em consideração essa realidade complexa, para então atender certos princípios contemporâneos, como o desenvolvimento urbano sustentável, reestruturação produtiva e os ambientes de pesquisa e inovação tecnológica e biotecnológica. Para tanto, agências de desenvolvimento foram criadas para conciliar os interesses públicos e privados envolvidos. Nesse contexto, utilizamos o conceito da cidade compacta, como base conceitual desta pesquisa à luz dos projetos urbanos contemporâneos em vazios urbanos em áreas centrais. Para contextualizar, buscamos os estudos de caso, construindo um inventário analítico das obras selecionadas, com maiores detalhamentos e sempre os relacionando à nossa realidade e contexto locais.

Esta investigación de tesis llamada Hibrido residencial en San Juan de Lurigancho responde a la gran demanda de vivienda por parte de las familias que conforman el distrito, buscando la calidad espacial dentro del hogar, la importancia de los espacios colectivos como dinamizadores sociales en un conjunto de viviendas o conjunto residencial, la necesidad de integrar en proyectos compuestos por muchas viviendas, equipamientos que los usuarios y el entorno necesiten y demostrar la importancia de la densificación en distritos con entornos consolidados para así poder lograr una ciudad compacta y dejar de usar indiscriminadamente el suelo. / This thesis research called Residential Hybrid in San Juan de Lurigancho, seeks to respond to the great demand for housing by the families that make up the district, without neglecting the spatial quality within the home, the importance of collective spaces as dynamizers social in a group of houses or residential complex, the need to integrate in projects composed of many houses, equipment that users and the environment need and demonstrate the importance of densification in districts with consolidated environments in order to achieve a compact city and leave to use the land indiscriminately. / Tesis

No description available.

Parkinson’s disease is an age-related neurodegenerative disorder. The major symptoms of Parkinson’s disease are closely linked to the pathology of the disease. The main pathology of Parkinson’s disease consists of the degeneration of neurons of the substantia nigra pars compacta (SNpc), which leads to reduced amounts of dopamine in the brain. One of the treatment strategies in Parkinson’s disease is to conserve dopamine by inhibiting the enzymes responsible for its catabolism. The monoamine oxidase (MAO) B isoform catalyses the oxidation of dopamine in the central nervous system and is therefore an important target for Parkinson’s disease treatment. Inhibition of MAO-B provides symptomatic relief for Parkinson’s disease patients by increasing endogenous dopamine levels as well as enhancing the levels of dopamine after administration of levodopa (L-dopa), the metabolic precursor of dopamine. Recent studies have shown that phthalide can be used as a scaffold for the design of reversible MAO inhibitors. Although phthalide is a weak MAO-B inhibitor, substitution on the C5 position of phthalide yields highly potent reversible MAO-B inhibitors. In the present study, sesamol and benzodioxane were used as scaffolds for the design of MAO inhibitors. The structures of sesamol and benzodioxane closely resemble that of phthalide, which suggests that these moieties may be useful for the design of MAO inhibitors. This study may be viewed as an exploratory study to discover new scaffolds for MAO inhibition. Since substitution at C5 of phthalide with a benzyloxy side chain yielded particularly potent MAO inhibitors, the sesamol and benzodioxane derivatives possessed the benzyloxy substituent in the analogous positions to C5 of phthalide. These were the C5 and C6 positions of sesamol and benzodioxane, respectively. The sesamol and benzodioxane derivatives were synthesised by reacting sesamol and 6- hydroxy-1,4-benzodioxane, respectively, with an appropriate alkyl bromide in the presence of potassium carbonate (K2CO3) in N,N-dimethylformamide (DMF). 6-Hydroxy-1,4- benzodioxane, in turn, was synthesised from 1,4-benzodioxan-6-carboxaldehyde. The structures of the compounds were verified with nuclear magnetic resonance (NMR) and mass spectrometry (MS) analyses, while the purities were estimated by high-pressure liquid chromatography (HPLC). Sixteen sesamol and benzodioxane derivatives were synthesised. To determine the inhibition potencies of the synthesised compounds the recombinant human MAO-A and MAO-B enzymes were used. The inhibition potencies were expressed as the corresponding IC50 values. The results showed that the sesamol and benzodioxane derivatives are highly potent and selective inhibitors of MAO-B and to a lesser extent MAOA. The most potent MAO-B inhibitor was 6-(3-bromobenzyloxy)-1,4-benzodioxane with an IC50 value of 0.045 μM. All compounds examined displayed selectivity for the MAO-B isoform over MAO-A. Generally the benzodioxane derivatives were found to be more potent inhibitors of human MAO-A and MAO-B than the sesamol derivatives. The reversibility and mode of MAO-B inhibition of a representative derivative, 6-(3- bromobenzyloxy)-1,4-benzodioxane, was examined by measuring the degree to which the enzyme activity recovers after dialysis of enzyme-inhibitor complexes, while Lineweaver- Burk plots were constructed to determine whether the mode of inhibition is competitive. Since MAO-B activity is completely recovered after dialysis of enzyme-inhibitor mixtures, it was concluded that 6-(3-bromobenzyloxy)-1,4-benzodioxane binds reversibly to the MAO-B enzyme. The Lineweaver-Burk plots constructed were linear and intersected on the y-axis. Therefore it may be concluded that 6-(3-bromobenzyloxy)-1,4-benzodioxane is a competitive MAO-B inhibitor. To conclude, the C6-substituted benzodioxane derivatives are potent, selective, reversible and competitive inhibitors of human MAO-B. These compounds are therefore promising leads for the future development of therapy for Parkinson’s disease. / MSc (Pharmaceutical Chemistry), North-West University, Potchefstroom Campus, 2015

Parkinson’s disease is an age-related neurodegenerative disorder. The major symptoms of Parkinson’s disease are closely linked to the pathology of the disease. The main pathology of Parkinson’s disease consists of the degeneration of neurons of the substantia nigra pars compacta (SNpc), which leads to reduced amounts of dopamine in the brain. One of the treatment strategies in Parkinson’s disease is to conserve dopamine by inhibiting the enzymes responsible for its catabolism. The monoamine oxidase (MAO) B isoform catalyses the oxidation of dopamine in the central nervous system and is therefore an important target for Parkinson’s disease treatment. Inhibition of MAO-B provides symptomatic relief for Parkinson’s disease patients by increasing endogenous dopamine levels as well as enhancing the levels of dopamine after administration of levodopa (L-dopa), the metabolic precursor of dopamine. Recent studies have shown that phthalide can be used as a scaffold for the design of reversible MAO inhibitors. Although phthalide is a weak MAO-B inhibitor, substitution on the C5 position of phthalide yields highly potent reversible MAO-B inhibitors. In the present study, sesamol and benzodioxane were used as scaffolds for the design of MAO inhibitors. The structures of sesamol and benzodioxane closely resemble that of phthalide, which suggests that these moieties may be useful for the design of MAO inhibitors. This study may be viewed as an exploratory study to discover new scaffolds for MAO inhibition. Since substitution at C5 of phthalide with a benzyloxy side chain yielded particularly potent MAO inhibitors, the sesamol and benzodioxane derivatives possessed the benzyloxy substituent in the analogous positions to C5 of phthalide. These were the C5 and C6 positions of sesamol and benzodioxane, respectively. The sesamol and benzodioxane derivatives were synthesised by reacting sesamol and 6- hydroxy-1,4-benzodioxane, respectively, with an appropriate alkyl bromide in the presence of potassium carbonate (K2CO3) in N,N-dimethylformamide (DMF). 6-Hydroxy-1,4- benzodioxane, in turn, was synthesised from 1,4-benzodioxan-6-carboxaldehyde. The structures of the compounds were verified with nuclear magnetic resonance (NMR) and mass spectrometry (MS) analyses, while the purities were estimated by high-pressure liquid chromatography (HPLC). Sixteen sesamol and benzodioxane derivatives were synthesised. To determine the inhibition potencies of the synthesised compounds the recombinant human MAO-A and MAO-B enzymes were used. The inhibition potencies were expressed as the corresponding IC50 values. The results showed that the sesamol and benzodioxane derivatives are highly potent and selective inhibitors of MAO-B and to a lesser extent MAOA. The most potent MAO-B inhibitor was 6-(3-bromobenzyloxy)-1,4-benzodioxane with an IC50 value of 0.045 μM. All compounds examined displayed selectivity for the MAO-B isoform over MAO-A. Generally the benzodioxane derivatives were found to be more potent inhibitors of human MAO-A and MAO-B than the sesamol derivatives. The reversibility and mode of MAO-B inhibition of a representative derivative, 6-(3- bromobenzyloxy)-1,4-benzodioxane, was examined by measuring the degree to which the enzyme activity recovers after dialysis of enzyme-inhibitor complexes, while Lineweaver- Burk plots were constructed to determine whether the mode of inhibition is competitive. Since MAO-B activity is completely recovered after dialysis of enzyme-inhibitor mixtures, it was concluded that 6-(3-bromobenzyloxy)-1,4-benzodioxane binds reversibly to the MAO-B enzyme. The Lineweaver-Burk plots constructed were linear and intersected on the y-axis. Therefore it may be concluded that 6-(3-bromobenzyloxy)-1,4-benzodioxane is a competitive MAO-B inhibitor. To conclude, the C6-substituted benzodioxane derivatives are potent, selective, reversible and competitive inhibitors of human MAO-B. These compounds are therefore promising leads for the future development of therapy for Parkinson’s disease. / MSc (Pharmaceutical Chemistry), North-West University, Potchefstroom Campus, 2015

Dopamine (DA) neurons of the substantia nigra pars compacta (SNc) have a key role in regulation of voluntary movement control. Their death is a hallmark of Parkinson’s disease, characterised by inhibited motor control, including muscle rigidity and tremor. Excitatory input to SNc-DA neurons is primarily from the subthalamic nucleus, and in PD these afferents display a higher frequency firing, as well as increased burst firing, which could cause increased excitatory activity in SNc-DA neurons. NMDA receptors (NMDARs) bind the excitatory neurotransmitter glutamate, and are essential for learning and memory. In SNc-DA neurons, NMDARs have a putative triheteromeric subunit arrangement of GluN1 plus GluN2B and/or GluN2D. Wild type (WT) mice, and those lacking the gene for GluN2D (Grin2D-null), were used to explore its role in various aspects of DA neuronal function and dysfunction using patch-clamp electrophysiology, viability assaying, and immunofluorescence. Pharmacological intervention using subunit-specific inhibitors ifenprodil and DQP-1105 on elicited NMDAR-EPSCs suggested a developmental shift from primarily GluN2B to GluN2B/D. Activity dependent regulation was assessed by high frequency burst stimulation of glutamatergic afferents: in comparison to controls, significant downregulation of NMDARs was observed in SNc-DA neurons, though no differences were observed based on genotype. This regulatory function may be a neuroprotective or homeostatic response. Ambient extracellular glutamate elicits tonic NMDAR activity in SNc-DA neurons, which may be important for maintaining basal levels of excitability: the role of GluN2D was assessed by recording the deflection in baseline current caused by application of competitive NMDAR antagonist D-AP5. There was a significantly larger NMDAR-mediated current in WT vs Grin2D-null mice, indicating that GluN2D has a role in binding ambient glutamate. Dysfunction of glutamate uptake could be a secondary pathophysiological occurrence in the SNc, leading to increased ambient glutamate: the effect of this was explored by application of the competitive glutamate transporter blocker TBOA. Here, the NMDAR-mediated portion of this current was significantly higher in WT mice in comparison to Grin2D-null. Interestingly, dose-response data obtained from bath application of NMDA showed significantly larger currents in Grin2D-null animals vs WT, but only at the top of the response curve (~1-10 mM), which may indicate a capability for larger conductance in Grin2D-null animals at high NMDAR saturation due to replacement of GluN2D with GluN2B. GluN2D may therefore be neuroprotective, by attenuating peak current flow in response to very high agonist concentrations. Lastly, GluN2D has been found to decrease NMDAR open probability under hypoxic conditions, potentially conferring resistance to hypoxia / ischemia related excitotoxicity. Therefore, low (15% O2 / 80% N2 / 5% CO2) vs high (95% O2 / 5% CO2) oxygen conditions were used along with immunofluorescent propidium iodide cell death assaying and immunofluorescent labeling for DA neurons in order to compare levels of DA neuronal death in the SNc based on oxygen status and genotype. Whilst there was a significant submaximal effect based on O2 status, genotype did not confer a practical resistance under these conditions. In summary, NMDARs have diverse roles in SNc-DA neurons which may both serve to maintain normal function and protect the cell against potentially pathological conditions.

The goal of this dissertation is to present a R. Reilly's demonstration of the theorem of Alexandrov . The theorem states that The only compact hypersurfaces, conected, of constant mean curvature, immersed in Euclidean space are spheres. The theorem of Alexandrov was proved by A. D. Alexandrov in the article Uniqueness Theorems for Surfaces in the Large V, published in 1958 by Vestnik Leningrad University, volume 13, number 19, pages 5 to 8. In his demonstration, Alexandrov used the famous Principle of tangency, introduced by him in that article. In the year 1962, M. Obata shown in Certain Conditions for a Riemannian Manifold to be isometric With the Sphere, published by the Journal of Mathematical Society of Japan, volume 14, pages 333 to 340, that a Riemannian Manifold M, compact, connected and without boundary, is isometric to a sphere, since the Ricci curvature of M satisfies certain lower bound. This theorem solves the problem of finding manifolds that reach equality in the estimate of Lichnerowicz for the first eigenvalue. In 1977, R. Reilly, in the article Applications of the Hessian operator in a Riemannian Manifold, published in Indianna University Mathematical Journal, volume 23, pages 459 to 452, showed a generalization of the Obata theorem for compact manifolds with boundary. As an example of the technique developed in this demonstration, he presents a new demonstration of the theorem of Alexandrov. This demonstration, as well as the techniques involved are the object of study of this work. / Conselho Nacional de Desenvolvimento Científico e Tecnológico / O objetivo desta dissertação é apresentar uma demonstração de R. Reilly para o Teorema de Alexandrov. O teorema estabelece que As únicas hipersuperfícies compactas, conexas, de curvatura média constante, mergulhadas no espaço Euclidiano são as esferas. O teorema de Alexandrov foi provado por A. D. Alexandrov no artigo Uniqueness Theorems for Surfaces in the Large V, publicado em 1958 pela Vestnik Leningrad University, volume 13, número 19, páginas 5 a 8. Em sua demonstração, Alexandrov usou o famoso Princípio de Tangência, introduzido por ele no citado artigo. No ano de 1962, M. Obata demonstrou em Certain Conditions for a Riemannian Manifold to be Isometric With a Sphere, publicado pelo Journal of Mathematical Society of Japan, volume 14, páginas 333 a 340, que uma variedade Riemanniana M, compacta, conexa e sem bordo, é isométrica a uma esfera, desde que a curvatura de Ricci de M satisfaça determinada limitação inferior. Este teorema resolve o problema de encontrar as variedades que atingem a igualdade na estimativa de Lichnerowicz para o primeiro autovalor. Em 1977, R. Reilly, no artigo Applications of the Hessian Operator in a Riemannian Manifold, publicado no Indianna University Mathematical Journal, volume 23, páginas 459 a 452, demonstrou uma generalização do Teorema de Obata para variedades compactas com bordo. Como exemplo da técnica desenvolvida nesta demonstração, ele apresenta uma nova demonstração do Teorema de Alexandrov. Esta demonstração, bem como as técnicas envolvidas, são o objeto de estudo deste trabalho.