The principal aim of this paper is to give an example of the Gauss-Bonnet Theorem together with its anew structure by using connection and curvature matrices with stereographic projection on the unit 2-sphere, S². We determine an orthonormal basis by applying stereographic projection on S² and we obtain the area of the unit 2 -sphere S² computing connection and curvature matrices.

From fundamental forms to curvatures and geodesics, differential geometry has many special theorems and applications worth examining. Among these, the Gauss-Bonnet Theorem is one of the well-known theorems in classical differential geometry. It links geometrical and topological properties of a surface. The thesis introduced some basic concepts in differential geometry, explained them with examples, analyzed the Gauss-Bonnet Theorem and presented the proof of the theorem in greater detail. The thesis also considered applications of the Gauss-Bonnet theorem to some special surfaces.

Habilitationsschrift, Bonn, 1971; extra t.p. inserted. / Includes bibliographical references (p. [92-93]).

Habilitationsschrift, Bonn, 1971; extra t.p. inserted. / Includes bibliographical references (p. [92-93]).

No description available.

RODRIGUES, Diego de Sousa. Desigualdades de Hitchin-Thorpe e Miyaoka-Yau. 2014. 55 f. Dissertação (Mestrado em Matemática) - Centro de Ciências, Universidade Federal do Ceará, Fortaleza, 2014. / Submitted by Erivan Almeida (eneiro@bol.com.br) on 2015-09-08T16:37:21Z No. of bitstreams: 1 2014_dis_dsrodrigues.pdf: 1115564 bytes, checksum: bbc98510dd8517874ebda43efb7b70b2 (MD5) / Approved for entry into archive by Rocilda Sales(rocilda@ufc.br) on 2015-09-09T11:45:06Z (GMT) No. of bitstreams: 1 2014_dis_dsrodrigues.pdf: 1115564 bytes, checksum: bbc98510dd8517874ebda43efb7b70b2 (MD5) / Made available in DSpace on 2015-09-09T11:45:06Z (GMT). No. of bitstreams: 1 2014_dis_dsrodrigues.pdf: 1115564 bytes, checksum: bbc98510dd8517874ebda43efb7b70b2 (MD5) Previous issue date: 2014 / The aim of this work is to present a proof of the Hitchin-Thorpe and Miyaoka-Yau inequalities. First we provide an orthogonal decomposition for the curvature tensor, and then we show how the curvature operator can be defined from the curvature tensor. In order to fulfill the proposed objective, we prove the Gauss-Bonnet Theorem in dimension 4, to do this we use a result due Allendoerfer and we present an integral formula for the Euler characteristic computation on a Riemannian 4-manifold. Furthermore, we define the concept of signature in a Riemannian manifold e we exhibit an integral formula for the achievement of this object, for this we use the Hirzebruch Signature Theorem in di- mension 4 and the Chern-Weil Theory which provides us a connection between algebraic topology and differential geometry. Finally, we show how the earlier formulas can be used in the demonstration of the initial inequalities. / O objetivo desse trabalho é fornecer uma demonstraçao para as desigualdades de Hitchin-Thorpe e Miyaoka-Yau. Inicialmente forneceremos uma decomposição ortogonal para o tensor curvatura, em seguida mostraremos como o operador curvatura pode ser definido a partir do tensor curvatura. Com o intuito de cumprir o objetivo proposto, iremos provar o Teorema de Gauss-Bonnet em dimensão 4, para isso utilizaremos um resultado devido a Allendoerfer e forneceremos uma fórmula integral para o cálculo da característica de Euler de uma variedade Riemanniana de dimensão 4. Além disso, definiremos o conceito de assinatura em uma variedade Riemanniana e exibiremos uma fórmula integral para a obtenção deste objeto, para isso utilizaremos o Teorema de Assinatura de Hirzebruch em dimensão 4 e pouco da Teoria de Chern-Weil que nos fornece uma conexão entre a topologia algébrica e a geometria diferencial. Por fim, mostraremos como as fórmulas que foram obtidas podem ser utilizadas na demonstraçao das desigualdades citadas inicialmente.

Nesse trabalho, estudamos o método do referencial móvel e sistemas diferenciais exteriores. Estabelecemos resultados de Geometria Riemanniana via referenciais móveis e com essa linguagem introduzimos o Teorema de Gauss-Bonnet-Chern e apresentamos uma adaptação da demonstração original de S.-S. Chern presente no artigo A simple intrinsic proof of the Gauss-Bonnet formula for closed Riemannian manifolds. Ao abordar aspectos da teoria de Cartan-Kähler, codificamos as ideias oriundas dos referenciais móveis em sistemas diferenciais exteriores e mostramos algumas aplicações à Geometria Riemanniana. / In this work, we study the method of moving frame and exterior differential systems. We set up results of Riemannian Geometry via moving frames and with this language we introduce the Gauss-Bonnet-Chern Theorem and present an adaptation of the original proof of S.-S. Chern in the article A simple intrinsic proof of the Gauss-Bonnet formula for closed Riemannian manifolds. In discussing aspects of Cartan-Kählers theory, we encode the ideas from moving frames into exterior differential systems and use this tool in Riemannian Geometry.

Στην εργασία αυτή θα παρουσιάσουμε το θεώρημα Gauss-Bonnet. Το θεώρημα αυτό είναι ένα από τα σημαντικότερα θεωρήματα της θεωρίας επιφανειών. Για πρώτη φορά δημοσιεύθηκε από τον O. Bonnet (1819-1892) το 1848, αλλά πιθανότατα να ήταν γνωστό στον Gauss. Μελετάμε το ολοκλήρωμα της καμπυλότητας Gauss K μιας συμπαγούς προσανατολισμένης επιφάνειας S. Στη συνέχεια δείχνουμε τη συσχέτιση του ολοκληρώματος αυτού, με την χαρακτηριστική του Euler, η οποία είναι μια σημαντική τοπολογική αναλλοίωτος της επιφάνειας S. Επίσης αναφερόμαστε στη γενίκευση του θεωρήματος Gauss-Bonnet σε μεγαλύτερες διαστάσεις. / In this work we study the Gauss-Bonnet Theorem. This theorem is one of the most important theorems in differential geometry of surfaces. Ιt was published by O. Bonnet (1819-1892) in 1848, but propably it was also known to Gauss. We study the integral of the Gauss curvature K of a compact, orientable surface S. Next we describe the connection of this integral with the Euler characteristic which is an important topological invariant of S. We also exam the generalization of the Gauss-Bonnet theorem in bigger dimensions.

A la tesi doctoral amb títol "Geometria integral en espais de curvatura holomorfa constant" es resolen qüestions de geometria integral clàssica però en espais de curvatura holomorfa constant, és a dir, a l'espai hermític estàndard, a l'espai projectiu complex i a l'espai hiperbòlic complex. Per assolir l'objectiu, primer de tot, es resumeixen les principals propietats i definicions de varietats de Kähler i, en particular, dels espais de curvatura holomorfa constant. També s'introdueix el concepte de valoració en espais vectorials. Una valoració és un funcional a valors reals, de l'espai de dominis convexos, compactes, no buits, que satisfan una propietat d'additivitat. Aquest concepte està a la base de quasi tots els resultats d'aquest treball ja que aquesta noció es pot estendre en varietats regulars. Així doncs, es dedica un capítol a definir els exemples de valoracions que s'utilitzaran i també a descriure noves propietats (variacionals) de les valoracions en els espais de curvatura holomorfa constant. En aquest punt, es donen els principals resultats de la tesis. Un dels problemes d'estudi de la geometria integral clàssica consisteix a donar una expressió de la mesura de plans que talla un domini fixat de l'espai euclidià, en termes de la geometria del domini. La fórmula que s'obté a l'espai euclidià involucra els volums mixtos (o, equivalentment, per dominis amb frontera regular, les integrals de curvatura mitjana del domini). En els altres espais de curvatura seccional constant (és a dir, a l'espai projectiu i hiperbòlic real) també se satisfà una fórmula que involucra els volums mixtos. En aquest treball s'obté una expressió de la mesura de plans complexos (de dimensió complexa des de 1 fins a n − 1, on n és la dimensió complexa de l'espai ambient) que talla un domini compacte amb frontera regular. L'expressió s'obté en termes de les valoracions anomenades volums intrínsecs hermítics, que es defineixen al segon capítol de la tesis. Per provar la certesa d'aquesta expressió s'utilitzen noves fórmules variacionals, tant per la mesura de plans complexos que tallen un domini com pels volums intrínsecs hermítics. A partir del mètode variacional anterior, s'obté la fórmula de Gauss-Bonnet-Chern a l'espai projectiu i hiperbòlic complexos. A més a més, es relaciona la característica d'Euler d'un domini compacte amb la mesura d'hiperplans complexos que tallen el domini i la integral de la curvatura de Gauss. Per altra banda, s'estudia la propietat de reproductibilitat de les integrals de curvatura mitjana. Als espais de curvatura seccional constant es té una propietat reproductiva, és a dir, la integral sobre l'espai de plans d'una integral de curvatura mitjana del domini intersecció és un m ́ultiple de la mateixa integral de curvatura mitja de tot el domini. En els espais de curvatura holomorfa constant aquesta propietat no es conserva. Aquest fet s'explica també a partir de la teoria de valoracions. La demostració involucra tècniques de geometria Riemanniana i referències mòbils. Finalment, es dóna la mesura de plans coisotròpics que tallen un domini a l'espai complex. S'anomena pla coisotròpic a aquell que el seu ortogonal és totalment real. També s'estudien propietats de les hipersuperfícies (reals) generades per l'exponencial en un punt (que no són totalment geodèsiques), sobre l'espai hiperbòlic complex. / The main goal of this work is to solve questions in classical integral geometry but for complex space forms, i.e. in the standard Hermitian space, the complex projective space and the complex hyperbolic space.In order to attain this goal, first of all, I survey the main properties and definitions concerning Kähler manifolds and, in particular, complex space forms. I also recall the notion of valuation in vector spaces. A valuation is a real-valued functional from the space of non-empty compact convex sets, satisfying an additive property. This notion is one of the main tools in this work since it can be extended to smooth manifolds. So, a chapter is devoted to the study of this notion and to describe new (variational) properties of some valuations in complex space forms. Then, the main results are stated. One of the problems of study of the classical integral geometry consists on giving an expression for the measure of planes meeting a domain in the Euclidean space, in terms of the geometry of the domain. The obtained formula in the Euclidean space involves the so-called intrinsic volumes (or equivalently for domains with regular boundary, the mean curvature integrals of the domain). In the other spaces with constant sectional curvature (i.e. in projective and hyperbolic space) it is also verified a formula involving the intrinsic volumes. In this work, I obtain an expression for the measure of complex planes (of complex dimension from 1 to n − 1, where n denotes the dimension of the ambient space) meeting a regular domain. The obtained expression is given in terms of the so-called Hermitian intrinsic volumes valuations, already defined at the second chapter. In order to prove this equality I use new variational formulas for the measure of complex planes intersecting a regular domain and for the Hermitian intrinsic volumes. From this variational method, I also get the Gauss-Bonnet-Chern formula in the complex projective and hyperbolic space. Moreover, I relate the Euler characteristic of a compact domain with the measure of complex hyperplanes meeting a compact domain, and the Gauss curvature. On the other hand, I study the reproductive property of the mean curvature integrals. In the spaces with constant sectional curvature, it is satisfied a reproductive property, i.e. the integral over the space of planes meeting a regular domain of the intersection domain is a multiple of the same mean curvature integral of the whole domain. In complex space forms this property it is not satisfied. This fact it is explained from the theory of valuations, and the proof involves techniques in Riemannian geometry and moving frames. Finally, I give the measure of coisotropic planes meeting a domain in the standard Hermitian space. A plane is called coisotropic if its orthogonal is totally real. I also study properties of the (real) hypersurfaces in complex hyperbolic space generated by the exponential map in a point, which are not totally geodesics.

In this dissertation, we investigatethe supersymmetric completion of curvature squared invariants in five and six dimensionsas well as the construction of off-shell Poincar´e supergravities and their matter couplings. We use superconformal calculus in fiveand six dimensions, which are an off- shell formalisms. In fivedimensions,there are twoinequivalentWeyl multiplets: the standard Weyl multiplet and the dilaton Weyl multiplet.The main difference betweenthese twoWeyl multiplets is thatthe dilaton Weyl multipletcontains a graviphoton in its field content whereas the standard Weyl multiplet does not.A supergravity theory based on the standard Weyl multiplet requires coupling to an external vector multiplet. In five dimensions,we construct two new formulations for 2-derivative off-shell Poincar´e supergravity theories and present the internally gauged models. We also construct supersymmetric completions of all curvature squared terms in five dimensional supergravity with eight supercharges.Adopting the dilaton Weyl multiplet, we construct a Weyl squared invariant, the supersymmetric combination of Gauss-Bonnet combination and the Ricci scalar squared invariant as well as all vector multiplets coupled curvature squared invariants. Since the minimal off-shell supersymmetric Riemann tensor squared invariant has been obtained before, both the minimal off-shell and the vector multiplets coupled curvature squared invariants in the dilation Weyl multiplet are complete. We also constructedan off-shell Ricci scalar squared invariant utilizing the standard Weyl multiplet.The supersymmetric Ricci scalar squared in the standard Weyl multiplet is coupled to n number of vector multiplets by construction, and it deforms the very special geometry. We found that in the supersymmetric AdS5 vacuum, the very special geometry defined on the moduli space is modified in a simple way. We study the vacuum solutions with AdS2 × S3 and AdS3 × S2 structures. We also analyze the spectrum around a maximally supersymmetric Minkowski5, and study the magnetic string and electric black hole. Finally, we generalize our procedure for the construction of an off-shell Ricci scalar squared invariant in five dimensions to N = (1, 0), D = 6 supergravity.