Σκοπός της εργασίας μας είναι η μελέτη κάποιων αναγωγικών χώρων που παρουσιάζουν ενδιαφέρουσα γεωμετρία. Συγκεκριμένα, μελετάμε τη γεωμετρία της σφαίρας S^n όταν αυτή είναι αμφιδιαφορική με έναν χώρο πηλίκο G/K και την γεωμετρία των πολλαπλοτήτων Stiefel SO(n)/SO(n-k) (το σύνολο όλων των k-πλαισίων του R^n). Ένας ομογενής χώρος αποτελεί επέκταση των ομάδων Lie, καθώς είναι μια λεία πολλαπλότητα M στην οποία δρα μεταβατικά μια ομάδα Lie G. Κάθε τέτοιος χώρος δίνεται ως M = G/K, όπου K = {g\in G : gp = p} (p \in M). Η βασική γεωμετρική ιδιότητα των ομογενών χώρων είναι ότι αν γνωρίζουμε την τιμή κάποιου γεωμετρικού μεγέθους σε ένα σημείο του χώρου, τότε μπορούμε να υπολογίσουμε την τιμή του μεγέθους αυτού σε οποιοδήποτε άλλο σημείο. Το ιδιαίτερο χαρακτηριστικό των αναγωγικών χώρων G/K είναι ότι υπάρχει ένας Ad(K)-αναλλοίωτος υπόχωρος της άλγεβρας Lie(G). Η περιγραφή όλων των μεταβατικών δράσεων μιας ομάδας Lie σε μια πολλαπλότητα M αποτελεί ένα δύσκολο πρόβλημα. Για την περίπτωση των σφαιρών αυτές έχουν περιγραφτεί το 1953 από τους Montgomery-Samelson-Borel. Στην εργασία μας μελετάμε τη γεωμετρία (καμπυλότητες, μετρικές Einstein) των σφαιρών S^3, S^5 όταν αυτές είναι αμφιδιαφορικές με τα πηλίκα S^3 = SO(4)/SO(3) = SU(2) και S^5 = SO(6)/SO(5) = SU(3)/SU(2). Αντίστοιχα προβλήματα εξετάζονται για τις πολλαπλότητες Stiefel SO(n)/SO(n-k), όπου η περιγραφή όλων των SO(n)-αναλλοίωτων μετρικών παρουσιάζει δυσκολία, λόγω του ότι η ισοτροπική αναπαράστασή τους περιέχει ισοδύναμα υποπρότυπα. Μελετάμε για ποιές από τις συγκεκριμένες πολλαπλότητες η μετρική που επάγεται από τη μορφή Killing είναι μετρική Einstein και περιγράφουμε αναλυτικά τις διαγώνιες SO(n)-αναλλοίωτες μετρικές Einstein στις πολλαπλότητες SO(n)/SO(n-2). Επιπλέον παρουσιάζουμε και ένα καινούργιο αποτέλεσμα, ότι στην πολλαπλότητα SO(5)/SO(2) οι μοναδικές SO(5)-αναλλοίωτες μετρικές Einstein είναι οι μετρικές που είχαν βρεθεί από τον Jensen το 1973. / The purpose of our work is to study homogeneous spaces that present interesting geometry. These include the geometry of the sphere S^n diffeomorphic to a quotient space G/K and the geometry of Stiefel manifolds SO(n)/SO(n-k) (the set of all k-planes in R^n). A homogeneous space is a smooth manifold M in which a Lie group acts transitively. Any such space is given as M = G/K where K = {g\in G : gp = p} (p\in M). The basic geometric property of homogeneous space is that if we know the value of a geometrical object at a point of the space, then we can estimate the value of thiw quantity at any other point. The special feature of reductive homogeneous space G/K is that there exists an Ad(K)-invariant subspace of the Lie algebra Lie(G). The description of all transitive actions of a Lie group into a manifold M is a difficult problem. In the case of spheres such actions have been described in 1953 by the Montgomery, Samelson and Borel. In our work we study the geometry (curvature, Einstein metrics) of the sphere S^3 = SO(4)/SO(3) = SU(2), S^5 = SO(6)/SO(5) = SU(3)/SU(2). Similar problems are examined for the Stiefel manifolds SO(n)/SO(n-k). The description of all SO(n)-invariant metrics presents serious difficulties because the isotropy representation contains equivalent submodules. We study for which of the manifolds SO(n)/SO(n-k) the metric induced by the Killing form is an Einstein metric and we describe in detail the diagonal SO(n)-invariant Einstein metrics on the Stiefel manifolds SO(n)/SO(n-2). In addition, we give the new result that for the Stiefel manifold SO(5)/SO(2) the unique SO(5)-invariant Einstein metrics are the metrics found by Jensen in 1973.

Les grassmanniennes symplectiques impaires sont une famille d'espaces quasi-homogènes très proches des grassmanniennes symplectiques de par leur construction et leurs propriétés. Dans ce travail, j'étudie leur cohomologie classique et quantique. Pour les grassmanniennes symplectiques impaires de droites, j'obtiens une règle de Pieri quantique ainsi qu'une présentation de l'anneau de cohomologie quantique. J'en déduis la semi-simplicité de cet anneau et je détermine une collection exceptionnelle complète pour la catégorie dérivée, ce qui me permet de vérifier pour cet exemple une conjecture de Dubrovin. Dans le cas général, je démontre un principe quantique-classique pour certains invariants de Gromov-Witten de degré un. Sous réserve de l'énumérativité des invariants de degré supérieur, je prouve que la règle de Pieri quantique est entièrement déterminée par le calcul des invariants de degré un. / Odd symplectic Grassmannians are a family of quasi-homogeneous spaces that are closely related to symplectic Grassmannians by their construction and properties. The goal of this work is to study their classical and quantum cohomology. For odd symplectic Grassmannians of lines, I obtain a quantum Pieri rule and a presentation of the quantum cohomology ring. I prove the semisimplicity of this ring and determine a full exceptional collection for the derived category, which enables me to check a conjecture of Dubrovin in this example. In the general case, I prove a quantum-to-classical principle for some degree one Gromov-Witten invariants. Assuming higher-dimensional Gromov-Witten invariants are enumerative, I conclude that the quantum Pieri rule is entirely determined by the knowledge of degree one invariants.

Neste trabalho analisamos a positividade definida estrita de núcleos contínuos sobre um espaço compacto 2-homogêneo. R. Gangolli (1967) apresentou uma caracterização completa para os núcleos que são contínuos, isotrópicos e positivos definidos sobre um espaço compacto 2-homogêneo Md: a parte isotrópica do núcleo é uma série de Fourier uniformemente convergente, com coeficientes não negativos, em relação a certos polinômios de Jacobi atrelados a Md. Uma das contribuições de nosso trabalho é uma caracterização para a positividade definida estrita de tais núcleos, complementando a caracterização apresentada por Chen et al. (2003) no caso em que Md é uma esfera unitária de dimensão maior ou igual a 2. Outra contribuição do trabalho é uma extensão do resultado de Gangolli para núcleos sobre produtos cartesianos de espaços compactos 2-homogêneos, e a consequente caracterização para núcleos estritamente positivos definidos neste mesmo contexto. Por fim, a última contribuição do trabalho envolve a análise do grau de diferenciabilidade da parte isotrópica de um núcleo contínuo, isotrópico e positivo definido sobre Md e a aplicabilidade de tal análise em resultados envolvendo a positividade definida estrita. / In this work we analyze the strict positive definiteness of continuous kernels on compact two-point homogeneous spaces Md. R. Gangolli (1967) presented a complete characterization for continuous, isotropic and positive definite kernels on Md: the isotropic part of the kernel is a uniformly convergent Fourier series of certain Jacobi polynomials associated to Md, with nonnegative coefficients. One of the contributions of our work is a characterization for the strict positive definiteness of such kernels, completing that one presented by Chen et al. (2003) in the case Md is the unit sphere of dimension at least 2. Another contribuition of this work is an extension of Gangolli\'s result for kernels on a product of compact two-point homogeneous spaces, and the subsequent characterization of strict positive definiteness in this same context. Finally, the last contribution in this work involves the analysis of the differentiability of the isotropic part of a continuous, isotropic and positive definite kernel on Md and the applicability of such analysis in results involving the strict positive definiteness.

<p>This thesis consists of two papers on the spectral geometry of locally symmetric spaces of Riemannian and Lorentzian signature. Both works are concerned with the idea of relating analysis on such spaces to structures on their boundaries.</p><p>The first paper is motivated by a conjecture of Patterson on the Selberg zeta function of Kleinian groups. We consider geometrically finite hyperbolic cylinders with non-compact Riemann surfaces of finite area as cross sections. For these cylinders, we present a detailed investigation of the Bunke-Olbrich extension operator under the assumption that the cross section of the cylinder has one cusp. We establish the meromorphic continuation of the extension of Eisenstein series and incomplete theta series through the limit set. Furthermore, we derive explicit formulas for the residues of the extension operator in terms of boundary values of automorphic eigenfunctions.</p><p>The motivation for the second paper comes from conformal geometry in Lorentzian signature. We prove the existence and uniqueness of a sequence of differential intertwining operators for spherical principal series representations, which are realized on boundaries of anti de Sitter spaces. Algebraically, these operators correspond to homomorphisms of generalized Verma modules. We relate these families to the asymptotics of eigenfunctions on anti de Sitter spaces.</p>

Ce mémoire porte sur l'étude des surfaces minimales et de courbure moyenne constante dans les espaces homogènes de dimension 3. Nous établissons les formules de Sym-Bobenko pour les surfaces de courbure moyenne constante 1/2 de H^2xR et minimales du groupe de Heisenberg, et donnons des exemples de construction de telles immersions par la méthode DPW. Nous montrons également que des propriétés de symétrie passent aux correspondances de type surfaces sœurs et cousines, ce qui entraîne l'existence de graphes entiers de courbure moyenne constante 1/2 à bout vertical dans H^2xR qui ne sont pas de révolution. Nous reprenons ensuite l'étude des bouts verticaux d'immersions de courbure moyenne constante 1/2 dans H^2xR. Nous munissons une famille de graphes entiers d'une structure de variété lisse et en déduisons un analogue pour H^2xR d'un théorème de A. E. Treibergs pour l'espace de Minkowski. Nous nous intéressons également aux déformations des anneaux de révolution. Une conséquence directe est l'existence d'anneaux immergés qui ne sont pas de révolution. Nous construisons notamment des anneaux dont les bouts n'ont pas le même axe. Enfin, nous décrivons les invariants de Nœther correspondant aux isométries des espaces homogènes pour les surfaces minimales et de courbure moyenne constante. Nous utilisons le formalisme de la géométrie de contact qui permet l'écriture de formules explicites en toute généralité, et nous étudions l'évolution des formes de Nœther sous l'action des isométries des espaces homogènes. Nous calculons ces invariants dans le cas des anneaux déformés de H^2xR, et dans celui des anneaux horizontaux du groupe de Heisenberg / The present dissertation deals with the study of minimal and constant mean curvature surfaces in 3-dimensional homogeneous spaces. In a first part, we establish Sym-Bobenko formulæ for constant mean curvature 1/2 surfaces in H^2xR and minimal surfaces in the Heisenberg group, and give examples of construction of such immersions using the DPW method. We also show that certain symmetry properties are shared by sister or cousin surfaces, which implies the existence non rotational entire graphs of constant mean curvature 1/2 in H^2xR with a vertical end.In a second part, we treat in more details the study of vertical ends of constant mean curvature 1/2 immersions in H^2xR. We endow a particular family entire graphs with a structure of smooth manifold and deduce an analogue in H^2xR to a theorem by A. E. Treibergs in the Minkowski space. We are also interested in deforming rotational annuli. A direct consequence is the existence of immersed non rotational annuli, and in particular we construct annuli with ends that do not have the same axis. Finally, we describe the Nœther invariants corresponding to isometries of the ambient homogeneous space for minimal and constant mean curvature surfaces. To do so, we use the formalism of contact geometry which allows general and explicit formulæ. We then study the evolution of Nœther form under the action of isometries in homogeneous spaces. We compute these invariants in the case of deformed annuli in H^2xR, and in the case of horizontal annuli in Heisenberg group

Este trabalho é composto de duas partes distintas, ambas dentro de um mesmo tema: núcleos positivos definidos sobre variedades. Na primeira delas fornecemos uma caracterização para os núcleos contínuos, isotrópicos e positivos definidos a valores matriciais sobre um espaço compacto 2-homogêneo. Utilizando-a, investigamos a positividade definida estrita destes núcleos, apresentando inicialmente algumas condições suficientes para garantir tal propriedade. No caso em que o espaço 2-homogêneo não é uma esfera, descrevemos uma caracterização definitiva para a positividade definida estrita do núcleo. Neste mesmo caso, para núcleos a valores no espaço das matrizes de ordem 2, apresentamos uma caraterização alternativa para a positividade definida estrita do núcleo via os dois elementos na diagonal principal da representação matricial do núcleo. Na segunda parte, nos restringimos a núcleos positivos definidos escalares sobre os mesmos espaços e determinamos condições necessárias e suficientes para a positividade definida estrita de um produto de núcleos positivos definidos sobre um mesmo espaço compacto 2-homogêneo. Apresentamos ainda uma extensão deste resultado para núcleos positivos definidos sobre o produto cartesiano de um grupo localmente compacto com uma esfera de dimensão alta, mantendo-se a isotropia na componente esférica. / In this work we present a characterization for the continuous, isotropic and positive definite matrix-valued kernels on a compact two-point homogeneous space. After that, we consider the strict positive definiteness of the kernels, describing some independent sufficient conditions for that property to hold. In the case the space is not a sphere, one of the conditions becomes necessary and sufficient for the strict positive definiteness of the kernel. Further, for 22- matrix-valued kernels on a compact two-point homogeneous space which is not a sphere, we present a characterization for the strict positive definiteness of the kernels based upon the main diagonal elements in its matrix representation. In the last part of this work, we restrict ourselves to scalar kernels and determine necessary and sufficient conditions in order that the product of two continuous, isotropic and positive definite kernels on a compact two-point homogeneous space be strictly positive definite. We also discuss the extension of this result for kernels defined on a product of a locally compact group and a high dimensional sphere.

This thesis consists of two papers on the spectral geometry of locally symmetric spaces of Riemannian and Lorentzian signature. Both works are concerned with the idea of relating analysis on such spaces to structures on their boundaries. The first paper is motivated by a conjecture of Patterson on the Selberg zeta function of Kleinian groups. We consider geometrically finite hyperbolic cylinders with non-compact Riemann surfaces of finite area as cross sections. For these cylinders, we present a detailed investigation of the Bunke-Olbrich extension operator under the assumption that the cross section of the cylinder has one cusp. We establish the meromorphic continuation of the extension of Eisenstein series and incomplete theta series through the limit set. Furthermore, we derive explicit formulas for the residues of the extension operator in terms of boundary values of automorphic eigenfunctions. The motivation for the second paper comes from conformal geometry in Lorentzian signature. We prove the existence and uniqueness of a sequence of differential intertwining operators for spherical principal series representations, which are realized on boundaries of anti de Sitter spaces. Algebraically, these operators correspond to homomorphisms of generalized Verma modules. We relate these families to the asymptotics of eigenfunctions on anti de Sitter spaces.

Μια πολλαπλότητα Riemann (M, g) ονομάζεται Einstein αν έχει σταθερή καμπυλότητα Ricci. Είναι γνωστό ότι αν (M=G/K, g) είναι μια συμπαγής ομογενής πολλαπλότητα Riemann, τότε οι G-αναλλοίωτες μετρικές Einstein μοναδιαίου όγκου, είναι τα κρίσιμα σημεία του συναρτησοειδούς ολικής βαθμωτής καμπυλότητας περιορισμένο στο χώρο των G-αναλλοίωτων μετρικών με όγκο 1. Για μια G-αναλλοίωτη μετρική Riemann η εξίσωση Einstein ανάγεται σε ένα σύστημα αλγεβρικών εξισώσεων. Οι θετικές πραγματικές λύσεις του συστήματος αυτού είναι ακριβώς οι G-αναλλοίωτες μετρικές Einstein που δέχεται η πολλαπλότητα Μ. Μια σημαντική οικογένεια συμπαγών ομογενών χώρων αποτελείται από τις γενικευμένες πολλαπλότητες σημαιών. Κάθε τέτοιος χώρος είναι μια τροχιά της συζυγούς αναπαράστασης μιας συμπαγούς, συνεκτικής, ημι-απλής ομάδας Lie G. Πρόκειται για ομογενείς πολλαπλότητες της μορφής G/C(S), όπου C(S) είναι ο κεντροποιητής ενός δακτυλίου S στην G. Κάθε τέτοιος χώρος δέχεται ένα πεπερασμένο πλήθος από G-αναλλοίωτες μετρικές Kahler-EInstein. Στην παρούσα διατριβή ταξινομούμε όλες τις πολλαπλότητες σημαιών G/K που αντιστοιχούν σε μια απλή ομάδα Lie G, των οποίων η ισοτροπική αναπαράσταση διασπάται σε 2 ή 4 μη αναγώγιμους και μη ισοδύναμους Ad(K)-αναλλοίωτους προσθετέους. Για κάθε τέτοιο χώρο λύνουμε την αναλλοίωτη εξίσωση Εinstein, και παρουσιάζουμε την αναλυτική μορφή νέων G-αναλλοίωτων μετρικών Einstein. Στις περισσότερες περιπτώσεις παρουσιάζουμε την πλήρη ταξινόμηση των αναλλοίωτων μετρικών Einstein. Επίσης εξετάζουμε το ισομετρικό πρόβλημα. Για την κατασκευή της εξίσωσης Einstein σε κάποιες πολλαπλότητες σημαιών με 4 ισοτροπικούς προσθετέους χρησιμοποιούμε την νηματοποίηση συστροφής που δέχεται κάθε πολλαπλότητα σημαιών επί ενός ισοτροπικά μη αναγώγιμου συμμετρικού χώρου συμπαγούς τύπου. Αυτή η μέθοδος είναι καινούργια και μπορεί να εφαρμοστεί και σε άλλες πολλαπλότητες σημαιών. / A Riemannian manifold (M, g) is called Einstein, if it has constant Ricci curvature. It is well known that if (M=G/K, g) is a compact homogeneous Riemannian manifold, then the G-invariant \tl{Einstein} metrics of unit volume, are the critical points of the scalar curvature function restricted to the space of all G-invariant metrics with volume 1. For a G-invariant Riemannian metric the Einstein equation reduces to a system of algebraic equations. The positive real solutions of this system are the $G$-invariant Einstein metrics on M. An important family of compact homogeneous spaces consists of the generalized flag manifolds. These are adjoint orbits of a compact semisimple Lie group. Flag manifolds of a compact connected semisimple Lie group exhaust all compact and simply connected homogeneous Kahler manifolds and are of the form G/C(S), where C(S) is the centralizer (in G) of a torus S in G. Such homogeneous spaces admit a finite number of G-invariant complex structures, and for any such complex structure there is a unique compatible G-invariant Kahler-Einstein metric. In this thesis we classify all flag manifolds M=G/K of a compact simple Lie group G, whose isotropy representation decomposes into 2 or 4, isotropy summands. For these spaces we solve the (homogeneous) Einstein equation, and we obtain the explicit form of new G-invariant Einstein metrics. For most cases we give the classification of homogeneous Einstein metrics. We also examine the isometric problem. For the construction of the Einstein equation on certain flag manifolds with four isotropy summands, we apply for first time the twistor fibration of a flag manifold over an isotropy irreducible symmetric space of compact type. This method is new and it can be used also for other flag manifolds. For flag manifolds with two isotropy summands, we use the restricted Hessian and we characterize the new Einstein metrics as local minimum points of the scalar curvature function restricted to the space of G-invariant Riemannian metrics of volume 1. We mention that the classification of flag manifolds with two isotropy summands gives us new examples of homogeneous spaces, for which the motion of a charged particle under the electromagnetic field, and the geodesics curves, are completely determined.

VASCONCELOS, Rosa Tayane de. O tensor de Ricci e campos de killing de espaços simétricos. 2017. 81 f. Dissertação (Mestrado em Matemática)- Centro de Ciências, Universidade Federal do Ceará, Fortaleza, 2017. / Submitted by Andrea Dantas (pgmat@mat.ufc.br) on 2017-09-18T13:45:50Z No. of bitstreams: 1 2017_dis_rtvasconcelos.pdf: 555452 bytes, checksum: 4ff6c8fb7950682913acabed03e9d3d7 (MD5) / Rejected by Rocilda Sales (rocilda@ufc.br), reason: Boa tarde, A Dissertação de ROSA TAYANE DE VASCONCELOS apresenta a alguns erros que devem corrigidos, os mesmos seguem listados abaixo: 1- EPÍGRAFE (coloque o nome do autor da epígrafe todo em letra maiúscula) 2- RESUMO/ ABSTRACT (retire o recuo dos parágrafos do resumo e do abstract) 3- PALAVRAS-CHAVE/ KEYWORDS (coloque a letra inicial do primeiro elemento das palavras- -chave e das Keywords em maiúscula) 4- CITAÇÕES (as citações a autores, que aparecem em todo o trabalho, não estão no padrão ABNT: se for apenas uma referência geral a uma obra, deve se colocar o último sobrenome do autor em letra maiúscula e o ano da publicação, ex.: EBERLEIN (2005). Caso seja a citação de um trecho particular da obra deve acrescentar o número da página, ex.: EBERLEIN (2005, p. 30). OBS.: as citações não devem estar entre colchetes. 5- TÍTULOS DOS CAPÍTULOS E SEÇÕES (coloque os títulos dos capítulos e seções em negrito) 6- REFERÊNCIAS (as referências bibliográficas não estão no padrão ABNT: apenas o último sobrenome do autor, que inicia a referência, deve estar em letra maiúscula, o restante do nome deve estar em letra minúscula. EX.: BROCKER, Theodor; TOM DIECK, Tammo. Representations of compact Lie groups, v. 98. Springer Science & Business Media, 2013. Atenciosamente, on 2017-09-18T15:04:06Z (GMT) / Submitted by Andrea Dantas (pgmat@mat.ufc.br) on 2017-09-19T13:33:40Z No. of bitstreams: 1 2017_dis_rtvasconcelos.pdf: 522079 bytes, checksum: ff99004fbe22e922f704a6a87365d3b6 (MD5) / Approved for entry into archive by Rocilda Sales (rocilda@ufc.br) on 2017-09-21T12:18:22Z (GMT) No. of bitstreams: 1 2017_dis_rtvasconcelos.pdf: 522079 bytes, checksum: ff99004fbe22e922f704a6a87365d3b6 (MD5) / Made available in DSpace on 2017-09-21T12:18:22Z (GMT). No. of bitstreams: 1 2017_dis_rtvasconcelos.pdf: 522079 bytes, checksum: ff99004fbe22e922f704a6a87365d3b6 (MD5) Previous issue date: 2017-09-13 / This work brings a smooth and self-contained introduction to the study of the most basic aspects of symmetric spaces, having as its nal goal the characterization of the Killing vector fields and of the Ricci tensor of such riemannian manifolds. Several of the results presented in the initial chapter are not easily found, in the Diferential Geometry literature, in a way as accessible and self-contained as here. This being said, we believe that this work embodies some didactic relevance, for it others students interested in symmetric spaces a relatively smooth first contact. We shall generally look at symmetric spaces as homogeneous manifolds G=H, where G is a Lie group and H is a closed Lie subgroup of G, such that the natural mapping : G ! G=H is a riemannian submersion. Ultimately, this map allows us to describe the relationships between the curvature, the Ricci tensor and the geodesics of G and G=H. For our purposes, the crucial remark is that, under appropriate circumstances, one guarantees the existence, in G=H, of a metric for which left translations are isometries. Hence, a one-parameter family of such isometries gives rise to a Killing vector field, which turn into a Jacobi vector eld when restricted to a geodesic. We present explicit expressions for such Jacobi vector elds, showing that they only depend on the eigenvalues of the linear operator TX : g ! g given by TX = (adX)2, for certain vector elds X 2 g. / Este trabalho traz uma introdução suave e autocontida ao estudo dos aspectos mais básicos de espaços simétricos, tendo como objetivo final a caracterização dos campos de Killing e do tensor de Ricci de tais variedades riemannianas. Vários dos resultados obtidos nos capítulos iniciais não são encontrados, na literatura de Geometria Diferencial, de maneira tão acessível e autocontida como apresentados aqui. Com isso, acreditamos que o trabalho reveste-se de alguma relevância didática, por oferecer aos alunos interessados no estudo de espaços simétricos um primeiro contato relativamente suave. Em linhas gerais, veremos espaços simétricos como variedades homogêneas G=H, onde G e um grupo de Lie e H um subgrupo de Lie fechado de G, tais que a aplicação natural: G ! G=H seja uma submersão riemanniana. Através dela, descrevemos relações entre a curvatura, o tensor de Ricci e as geodésicas de G e G=H. Para nossos propósitos, a observação crucial e que, sob certas hipóteses, garantimos a existência, em G=H, de uma métrica cujas translações a esquerda são isometrias. Portanto, uma família a um parâmetro de tais isometrias d a origem a um campo de Killing que, por sua vez, restrito a geodésicas torna-se um campo de Jacobi. Apresentamos expressões para esses campos de Jacobi, mostrando que os mesmos só dependem dos autovalores do operador linear TX : g ! g dado por TX = (adX)2, para certos campos X 2 g.

Este trabalho é composto de duas partes distintas, ambas dentro de um mesmo tema: núcleos positivos definidos sobre variedades. Na primeira delas fornecemos uma caracterização para os núcleos contínuos, isotrópicos e positivos definidos a valores matriciais sobre um espaço compacto 2-homogêneo. Utilizando-a, investigamos a positividade definida estrita destes núcleos, apresentando inicialmente algumas condições suficientes para garantir tal propriedade. No caso em que o espaço 2-homogêneo não é uma esfera, descrevemos uma caracterização definitiva para a positividade definida estrita do núcleo. Neste mesmo caso, para núcleos a valores no espaço das matrizes de ordem 2, apresentamos uma caraterização alternativa para a positividade definida estrita do núcleo via os dois elementos na diagonal principal da representação matricial do núcleo. Na segunda parte, nos restringimos a núcleos positivos definidos escalares sobre os mesmos espaços e determinamos condições necessárias e suficientes para a positividade definida estrita de um produto de núcleos positivos definidos sobre um mesmo espaço compacto 2-homogêneo. Apresentamos ainda uma extensão deste resultado para núcleos positivos definidos sobre o produto cartesiano de um grupo localmente compacto com uma esfera de dimensão alta, mantendo-se a isotropia na componente esférica. / In this work we present a characterization for the continuous, isotropic and positive definite matrix-valued kernels on a compact two-point homogeneous space. After that, we consider the strict positive definiteness of the kernels, describing some independent sufficient conditions for that property to hold. In the case the space is not a sphere, one of the conditions becomes necessary and sufficient for the strict positive definiteness of the kernel. Further, for 22- matrix-valued kernels on a compact two-point homogeneous space which is not a sphere, we present a characterization for the strict positive definiteness of the kernels based upon the main diagonal elements in its matrix representation. In the last part of this work, we restrict ourselves to scalar kernels and determine necessary and sufficient conditions in order that the product of two continuous, isotropic and positive definite kernels on a compact two-point homogeneous space be strictly positive definite. We also discuss the extension of this result for kernels defined on a product of a locally compact group and a high dimensional sphere.