Αντικείμενο της εργασίας αυτής είναι η αναζήτηση των διαρμονικών υποπολλαπλοτήτων της σφαίρας S3. Η μέθοδος που εφαρμόζεται συνδέεται με την αρχή του λογισμού των μεταβολών. Γίνεται σύντομη ανάλυση της μεθοδολογίας του λογισμού των μεταβολών και εφαρμογή αυτής σε γνωστές θεωρίες μεταξύ των οποίων είναι οι αρμονικές και διαρμονικές απεικονίσεις. Ορίζουμε τις έννοιες των αρμονικών και διαρμονικών απεικονίσεων μεταξύ δύο πολλαπλοτήτων Riemann και δίνουμαι παραδείγματα τέτοιων απεικονίσεων. Τέλος, προσδιορίζουμαι τις διαρμονικές καμπύλες και τις διαρμονικές επιφάνειες της σφαίρας S3. Οι κεντρικές μας αναφορές είναι οι εργασίες : (1) Biharmonic submanifolds in spheres, Israel.J.Math.,130(2002), 109-123, των R.Caddeo, S. Montaldo και C .Oniciuic. (2) A report on harmonic maps, Bull. London Math. Soc. 10(1978), 1-68 των J. Eells και L.Lemaire. / The object of this project is the investigation of the biharmonic submanifolds of sphere S3. The method we apply is the variational method. We shortly analyse the method of variations and we describe some theorys as they derived by this method. Between those theorys are the harmonic and biharmonic maps. We define the notions of harmonic and biharmonic maps between two Riemannian manifolds and we introduce some examples. Finally, we allocate the biharmonic curves and surfaces of sphere S3. The central references are: (1) Biharmonic submanifolds in spheres, Israel.J.Math.,130(2002), 109-123, των R.Caddeo, S. Montaldo και C .Oniciuic. (2) A report on harmonic maps, Bull. London Math. Soc. 10(1978), 1-68 των J. Eells και L.Lemaire.

Thesis (Doctoral)--Universität Bonn, 1990. / Includes bibliographical references.

Thesis (Ph. D.)--University of Washington, 1999. / Vita. Includes bibliographical references (leaves [86]-88).

Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / Estudamos folheaÃÃes de formas espaciais por hipersuperfÃcies completas, sob certas condiÃÃes sobre as suas curvaturas mÃdias de ordem superior. Em particular, no espaÃo euclidiano obtemos um Teorema tipo-Bernstein para grÃficos cujas curvaturas mÃdia e escalar nÃo mudam de sinal (podendo ser nÃo constantes). NÃs tambÃm estabelecemos a nÃo existÃncia de folheaÃÃes da esfera padrÃo cujas folhas sÃo completas e tÃm curvatura escalar constante, alargando assim um teorema de Barbosa, Kenmotsu e Oshikiri. Para o caso mais geral de folheaÃÃes r-mÃnimas do espaÃo euclidiano, possivelmente com um conjunto singular, somos capazes de invocar um teorema de D. Ferus para dar condiÃÃes sob as quais as folhas nÃo-singulares sÃo folheadas por hiperplanos. / We study foliations of space forms by complete hypersurfaces, under some mild conditions on its higher order mean curvatures. In particular, in Euclidean space we obtain a Bernstein-type theorem for graphs whose mean and scalar curvature do not change sign but may otherwise be nonconstant. We also establish the nonexistence of foliations of the standard sphere whose leaves are complete and have constant scalar curvature, thus extending a theorem of Barbosa, Kenmotsu and Oshikiri. For the more general case of r-minimal foliations of the Euclidean space, possibly with a singular set, we are able to invoke a theorem of Ferus to give conditions under which the nonsigular leaves are foliated by hyperplanes.

Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / Neste trabalho, definimos as transformaÃÃes de Newton e provamos algumas propriedades relacionadas a elas. Fizemos um estudo sobre operador elÃptico e usamos isso para provar que dadas algumas condiÃÃes para a curvatura seccional de uma variedade riemanniana M, conseguimos majorar a funÃÃo altura (em modulo) de um grÃfico vertical compacto imerso em MxR. / In this paper, we define the transformations of Newton and prove some properties related to them. We did a study on elliptic operator and use it to prove that given some conditions for the sectional curvature of a riemannian manifold M,able function of increasing height (in modulus) of a graph vertical compact immersed in MXR.

Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / Este trabalho consiste em duas partes. Na primeira parte, estudaremos hipersuperfÃcies compactas sem bordo imersas no espaÃo Euclidiano com o quociente das curvaturas mÃdias anisotrÃpicas constante. Provaremos que tais hipersuperfÃcies sÃo pontos crÃticos para um problema variacional de preservar uma combinaÃÃo linear da (k; F)-Ãrea e do (n+1)-volume determinado por M. Demostraremos que a hipersuperfÃcie à (r; k; a; b)-estÃvel se, e somente se, a menos de translaÃÃo e homotetia, ela à a Wulff shape de F (veja SeÃÃo 2.1), sob algumas condiÃÃes acerca de a; b â R. Na segunda parte desse trabalho, obtemos outras caracterizaÃÃes para a Wulff shape envolvendo as curvaturas mÃdias anisotrÃpicas de ordem superior de uma hipersuperfÃ- cie M em Rn+1 e o conjunto W = Rn+1 -UpâM Tp. Os resultados sÃo obtidos para hipersuperfÃcies compactas nÃo convexas satisfazendo W ╪ Ã. / This work consists of two parts. In the first part we deal with a compact hypersurface without boundary immersed in to the Euclidean space with the quotient of anisotropic mean curvatures constant. Such a hypersurface is a critical point for the variational problem preserving a linear combination of the (k; F)-area and (n + 1)-volume enclosed by M. We show that it is (r; k; a; b)-stable if, and only if, up to translations and homotheties, it is the Wulff shape, under some assumptions on a; b â R. In the second part we obtain further characterizations for the Wulff shape involving the anisotropic mean curvatures of higher order of a hypersurface M in Rn+1 and the set W = Rn+1-UpâM Tp. Results are obtained for non-convex compact hypersurfaces satisfying W ╪ Ã.

Made available in DSpace on 2014-06-12T18:28:51Z (GMT). No. of bitstreams: 2 arquivo4264_1.pdf: 8529909 bytes, checksum: 0a958aa0c57b9a6b54d2a7542dbf9476 (MD5) license.txt: 1748 bytes, checksum: 8a4605be74aa9ea9d79846c1fba20a33 (MD5) Previous issue date: 2007 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Um blink é um grafo plano onde cada aresta ou é vermelha ou é verde. Um espaço 3D ou, simplesmente, um espaço é uma variedade 3-dimensional conexa, fechada e orientada. Neste trabalho exploramos pela primeira vez em maiores detalhes o fato de que todo blink induz um espaço e todo espaço é induzido por algum blink (na verdade por infinitos blinks). Qual o espaço de um triângulo verde? E de um quadrado vermelho? São iguais? Estas perguntas foram condensadas numa pergunta cuja busca pela resposta guiou em grande parte o trabalho desenvolvido: quais são todos os espaços induzidos por blinks pequenos (poucas arestas)? Nesta busca lançamos mão de um conjunto de ferramentas conhecidas: os blackboard framed links (BFL), os grupos de homologia, o invariante quântico de Witten-Reshetikhin-Turaev, as 3-gems e sua teoria de simplificação. Combinamos a estas ferramentas uma teoria nova de decomposição/composição de blinks e, com isso, conseguimos identificar todos os espaços induzidos por blinks de até 9 arestas (ou BFLs de até 9 cruzamentos). Além disso, o nosso esforço resultou também num programa interativo de computador chamado BLINK. Esperamos que ele se mostre útil no estudo de espaços e, em particular, na descoberta de novos invariantes que complementem o invariante quântico resolvendo as duas incertezas deixadas em aberto neste trabalho

The problem of embedding a manifold in Euclidean space is considered. Manifolds are introduced in Chapter I along with other basic definitions and examples. Chapter II contains a proof of the Regular Value Theorem along with the "Easy" Whitney Embedding Theorem. In Chapter III, vector bundles are introduced and some of their properties are discussed. Chapter IV introduces the Stiefel-Whitney classes and the four properties that characterize them. Finally, in Chapter V, the Stiefel-Whitney classes are used to produce a lower bound on the dimension of Euclidean space that is needed to embed real projective space.

In this thesis we study the problem of determining the possible cobordism types of r-fold self-intersection manifolds associated to self-transverse immersions f: M^{n-k} -> \R^n for certain values of n, k, and r. Namely, we study the double-point self-intersection manifolds of immersions M^{n+2} -> \R^{2n+2} and M^{n+4} -> \R^{2n+4}, focusing on the case when $n$ is even. In the case of self-transverse immersions f : M^{n+2} -> \R^{2n+2}, we see that when n is even the double-point self-intersection manifold is a boundary, which is a result originally due to Szucs. In the case of self-transverse immersions f : M^{n+4} -> \R^{2n+4}, we show than when n is even the double-point self-intersection manifold is either a boundary or cobordant to RP^2 x RP^2, which is a new result. We then show that for even n such that the binary expansion of n+4 contains 5 or more 1s, the double-point self-intersection manifold of a self-transverse immersion M^{n+4} -> \R^{2n+4} is necessarily a boundary. We also survey the case when n is odd. We also set up and study the complex versions of the above problems: self-transverse immersions f : M^{2k+2} -> \R^{4k+2} and f : M^{2k+4} -> \R^{4k+4} of stably complex manifolds with a given complex structure on the normal bundle of f$. In these cases, the double-point self-intersection manifold L associated to the immersion inherits a stably complex structure, and we attempt to determine which complex cobordism classes of stably complex manifolds may arise in this way. This is all new work. In the case of self-transverse complex immersions f : M^{2k+2} -> \R^{4k+2}, we show that the first normal Chern number of the double-point self-intersection manifold is a multiple of 2^{\lambda_{k+1}} for some integer \lambda_{k+1}, and provide upper and lower bounds for the value of \lambda_{k+1}. We also determine the exact value of \lambda_{k+1} in certain cases. In the case of self-transverse complex immersions f : M^{2k+4} -> \R^{4k+4}, we identify a large class of stably complex manifolds that may arise as the double-point self-intersection manifold of such an immersion and also identify a class of manifolds that may not. Additionally, in both cases we identify a necessary (and sometimes sufficient) condition for a stably complex manifold of the appropriate dimension to admit a complex immersion of the appropriate codimension.

This thesis is concerned principally with tangential geometry and the applications of these concepts to tangentially symplectic foliations. The subject of tangential geometry is still at an elementary stage. The author here systematises current concepts and results and extends them, leading to the definition of vertical connections and vertical G-structures. Tangentially symplectic foliations are then characterised in terms of vertical symplectic forms. Some significant particular cases are discussed.