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]).

Differential geometry is about space (a manifold) and a geometric structure on that space. In Riemann’s lecture (see [17]), he stated that “Thus arises the problem, to discover the matters of fact from which the measure-relations of space may be determined...”. It is key then to understand how manifolds differ from one another geometrically. The results of this dissertation concern how the geometry of a manifold changes when we alter metrical connections. We investigate how diverse geodesics are in different metrical connections. From this, we investigate a new class of metrical connections which are dependent on the class of smooth functions. Specifically, we fix a Riemannian metric and investigate the geometry of the manifold when we change the metrical connections associated with the fixed Riemannian metric. We measure the change in the Riemannian curvatures associated with this new class of metrical connections, and then give uniqueness and existence criterion for curvature of compact 2-manifolds. These results depend on the use of Hodge Theory and ultimately on the function f we choose to define a metrical connection.

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.

Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / O objetivo desse trabalho à fornecer uma demonstraÃao para as desigualdades de Hitchin-Thorpe e Miyaoka-Yau. Inicialmente forneceremos uma decomposiÃÃo 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 fÃrmula integral para o cÃlculo da caracterÃstica de Euler de uma variedade Riemanniana de dimensÃo 4. AlÃm disso, definiremos o conceito de assinatura em uma variedade Riemanniana e exibiremos uma fÃrmula integral para a obtenÃÃo 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 algÃbrica e a geometria diferencial. Por fim, mostraremos como as fÃrmulas que foram obtidas podem ser utilizadas na demonstraÃao das desigualdades citadas inicialmente. / 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.

Made available in DSpace on 2014-06-11T19:27:10Z (GMT). No. of bitstreams: 0 Previous issue date: 2011-09-21Bitstream added on 2014-06-13T19:47:36Z : No. of bitstreams: 1 batista_ra_me_rcla.pdf: 818880 bytes, checksum: 6293c2c753e3d0bd5a6900cfc890944f (MD5) / O principal objetivo deste trabalho é confeccionar um texto para alunos de gradua ção na área de Ciências Exatas e da Terra concernente ao estudo da Curvatura Gaussiana e Aplicação de Gauss, Superfícies Mínimas, Teorema Egregium de Gauss e o Teorema de Gauss- Bonnet para curvas simples fechadas / The main objective from this work is to make a text for students of graduation in the area of exact sciences and of the land concerning to the study of the Gaussian Curvature and the Gauss Map, Minimal Surfaces, Gauss's Theorem Egregium and the Gauss-Bonnet Theorem for Simple Closed Curves

We introduce the formalism of supersymmetric quantum mechanics, including super-symmetry charges,Z2-graded Hilbert spaces, the chirality operator and the Wittenindex. We show that there is a one to one correspondence of fermions and bosons forenergies different than zero, which implies that the Witten index measures the dif-ference of fermions and bosons at the ground state. We argue that the Witten indexis the index of an elliptic operator. Quantization of the supersymmetric non-linearsigma model shows that the Witten index equals the Euler characteristic of the un-derlying Riemannian manifold over which the theory is defined. Finally, the pathintegral representation of the Witten index is applied to derive the Gauss-Bonnettheorem. Apart from this we introduce elementary mathematical background in thesubjects of topological invariance, Riemannian manifolds and index theory / Vi introducucerar formalismen f ̈or supersymmetrisk kvantmekanik, d ̈aribland super-symmetryladdningar,Z2-graderade Hilbertrum, kiralitetsoperatorn och Wittenin-dexet. Vi visar att det r ̊ader en till en-korrespondens mellan fermioner och bosonervid energiniv ̊aer skillda fr ̊an noll, vilket medf ̈or att Wittenindexet m ̈ater skillnadeni antal fermioner och bosoner vid nolltillst ̊andet. Vi argumenterar f ̈or att Wittenin-dexet ̈ar indexet p ̊a en elliptisk operator. Kvantisering av den supersymmetriskaicke-linj ̈ara sigmamodellen visar att Wittenindexet ̈ar Eulerkarakteristiken f ̈or denunderliggande Riemannska m ̊angfald ̈over vilken teorin ̈ar definierad. Slutligenapplicerar vi v ̈agintegralrepresentationen av Wittenindexet f ̈or att h ̈arleda Gauss-Bonnets sats. Ut ̈over detta introduceras ocks ̊a grundl ̈aggande matematisk bakrundi ämnena topologisk invarians, Riemmanska m ̊angfalder och indexteori.

Orientador: João Peres Vieira / Banca: Eliris Cristina Rizziolli / Banca: Laércio Aparecido Lucas / Resumo: O principal objetivo deste trabalho é confeccionar um texto para alunos de gradua ção na área de Ciências Exatas e da Terra concernente ao estudo da Curvatura Gaussiana e Aplicação de Gauss, Superfícies Mínimas, Teorema Egregium de Gauss e o Teorema de Gauss- Bonnet para curvas simples fechadas / Abstract: The main objective from this work is to make a text for students of graduation in the area of exact sciences and of the land concerning to the study of the Gaussian Curvature and the Gauss Map, Minimal Surfaces, Gauss's Theorem Egregium and the Gauss-Bonnet Theorem for Simple Closed Curves / Mestre

Given a hyperbolic surface, the set of all closed geodesics whose length is minimal form a graph on the surface, in fact a so called fat graph, which we call the systolic graph. The central question that we study in this thesis is: which fat graphs are systolic graphs for some surface -we call such graphs admissible. This is motivated in part by the observation that we can naturally decompose the moduli space of hyperbolic surfaces based on the associated systolic graphs. A systolic graph has a metric on it, so that all cycles on the graph that correspond to geodesics are of the same length and all other cycles have length greater than these. This can be formulated as a simple condition in terms of equations and inequations for sums of lengths of edges. We call this combinatorial admissibility. Our first main result is that admissibility is equivalent to combinatorial admissibility. This is proved using properties of negative curvature, specifically that polygonal curves with long enough sides, in terms of a lower bound on the angles, are close to geodesics. Using the above result, it is easy to see that a subgraph of an admissible graph is admissible. Hence it suffices to characterize minimal non-admissible fat graphs. Another major result of this thesis is that there are infinitely many minimal non-admissible fat graphs (in contrast, for instance, to the classical result that there are only two minimal non-planar graphs).