Differentiable Manifolds: A First Course by Lawrence Conlon

By Lawrence Conlon

The fundamentals of differentiable manifolds, worldwide calculus, differential geometry, and comparable issues represent a center of data crucial for the 1st or moment yr graduate pupil getting ready for complex classes in differential topology and geometry. Differential Manifolds is a textual content designed to hide this fabric in a cautious and sufficiently specified demeanour, presupposing just a strong grounding more often than not topology, calculus, and sleek algebra. it truly is perfect for a whole yr Ph.D. qualifying path and sufficiently self contained for personal examine by means of non-specialists wishing to survey the subject. the topics of linearization, (re)integration, and international as opposed to neighborhood are emphasised time and again; extra beneficial properties contain a remedy of the weather of multivariable calculus, an exploration of package concept, and a different improvement of Lie thought than is ordinary in textbooks at this point. scholars, academics, and pros in arithmetic and mathematical physics should still locate this a such a lot stimulating and beneficial textual content.

Show description

By Lawrence Conlon

The fundamentals of differentiable manifolds, worldwide calculus, differential geometry, and comparable issues represent a center of data crucial for the 1st or moment yr graduate pupil getting ready for complex classes in differential topology and geometry. Differential Manifolds is a textual content designed to hide this fabric in a cautious and sufficiently specified demeanour, presupposing just a strong grounding more often than not topology, calculus, and sleek algebra. it truly is perfect for a whole yr Ph.D. qualifying path and sufficiently self contained for personal examine by means of non-specialists wishing to survey the subject. the topics of linearization, (re)integration, and international as opposed to neighborhood are emphasised time and again; extra beneficial properties contain a remedy of the weather of multivariable calculus, an exploration of package concept, and a different improvement of Lie thought than is ordinary in textbooks at this point. scholars, academics, and pros in arithmetic and mathematical physics should still locate this a such a lot stimulating and beneficial textual content.

Show description

Read Online or Download Differentiable Manifolds: A First Course PDF

Similar mathematics books

Multiparameter Eigenvalue Problems and Expansion Theorems

This ebook offers a self-contained remedy of 2 of the most difficulties of multiparameter spectral thought: the lifestyles of eigenvalues and the growth in sequence of eigenfunctions. the consequences are first received in summary Hilbert areas after which utilized to quintessential operators and differential operators.

Séminaire Bourbaki, Vol. 1, 1948-1951, Exp. 1-49

Desk of Contents

* 1 Henri Cartan Les travaux de Koszul, I (Lie algebra cohomology)
* 2 Claude Chabauty Le théorème de Minkowski-Hlawka (Minkowski-Hlawka theorem)
* three Claude Chevalley L'hypothèse de Riemann pour les corps de fonctions algébriques de caractéristique p, I, d'après Weil (local zeta-function)
* four Roger Godement Groupe complexe unimodulaire, I : Les représentations unitaires irréductibles du groupe complexe unimodulaire, d'après Gelfand et Neumark (representation concept of the advanced distinctive linear group)
* five Léo Kaloujnine Sur l. a. constitution de p-groupes de Sylow des groupes symétriques finis et de quelques généralisations infinies de ces groupes (Sylow theorems, symmetric teams, countless staff theory)
* 6. Pierre Samuel l. a. théorie des correspondances birationnelles selon Zariski (birational geometry)
* 7 Jean Braconnier Sur les suites de composition d'un groupe et los angeles travel des groupes d'automorphismes d'un groupe fini, d'après H. Wielandt (finite groups)
* eight Henri Cartan, Les travaux de Koszul, II (see 1)
* nine Claude Chevalley, L'hypothèse de Riemann pour les groupes de fonctions algébriques de caractéristique p, II,, d'après Weil (see 3)
* 10 Luc Gauthier, Théorie des correspondances birationnelles selon Zariski (see 6)
* eleven Laurent Schwartz, Sur un mémoire de Petrowsky : "Über das Cauchysche challenge für ein process linearer partieller Differentialgleichungen im gebiete nichtanalytischen Funktionen" (partial differential equations)
* 12 Henri Cartan, Les travaux de Koszul, III (see 1)
* thirteen Roger Godement, Groupe complexe unimodulaire, II : los angeles transformation de Fourier dans le groupe complexe unimodulaire à deux variables, d'après Gelfand et Neumark (see 4)
* 14 Marc Krasner, Les travaux récents de R. Brauer en théorie des groupes (finite groups)
* 15 Laurent Schwartz, Sur un deuxième mémoire de Petrowsky : "Über das Cauchysche challenge für approach von partiellen Differentialgleichungen" (see 11)
* sixteen André Weil Théorèmes fondamentaux de l. a. théorie des fonctions thêta, d'après des mémoires de Poincaré et Frobenius (theta functions)
* 17 André Blanchard, Groupes algébriques et équations différentielles linéaires, d'après E. Kolchin (differential Galois theory)
* 18 Jean Dieudonné, Géométrie des espaces algébriques homogènes, d'après W. L. Chow (algebraic geometry)
* 19 Roger Godement, Sommes keeps d'espaces de Hilbert, I (functional research, direct integrals)
* 20 Charles Pisot, Démonstration élémentaire du théorème des nombres premiers, d'après Selberg et Erdös (prime quantity theorem)
* 21 Georges Reeb, Propriétés des trajectoires de certains systèmes dynamiques (dynamical systems)
* 22 Pierre Samuel, Anneaux locaux ; creation à l. a. géométrie algébrique (local rings)
* 23 Marie-Hélène Schwartz, Compte-rendu de travaux de M. Heins sur diverses majorations de l. a. croissance des fonctions analytiques et sous-harmoniques (complex research, subharmonic functions)
* 24 Charles Ehresmann, Les connexions infinitésimales dans un espace fibré différentiable (connections on fiber bundles)
* 25 Roger Godement, Sommes keeps d'espaces de Hilbert, II (see 19)
* 26 Laurent Schwartz, Sur un mémoire de ok. Kodaira : "Harmonic fields in riemannian manifolds (generalized capability theory)", I (Hodge theory)
* 27 Jean-Pierre Serre, Extensions de groupes localement compacts, d'après Iwasawa et Gleason (locally compact groups)
* 28 René Thom, Les géodésiques dans les variétés à courbure négative, d'après Hopf (geodesics)
* 29 Armand Borel, Groupes localement compacts, d'après Iwasawa et Gleason (see 27)
* 30 Jacques Dixmier, Facteurs : class, measurement, hint (von Neumann algebras)
* 31 Jean-Louis Koszul, Algèbres de Jordan (Jordan algebras)
* 32 Laurent Schwartz, Sur un mémoire de okay. Kodaira : "Harmonic fields in riemannian manifolds (generalized strength theory)", II (see 26)
* 33 Armand Borel, Sous-groupes compacts maximaux des groupes de Lie, d'après Cartan, Iwasawa et Mostow (maximal compact subgroups)
* 34 Henri Cartan, Espaces fibrés analytiques complexes (analytic geometry, fiber bundles)
* 35 Charles Ehresmann, Sur les variétés presque complexes (almost-complex manifolds)
* 36 Samuel Eilenberg, Exposition des théories de Morse et Lusternick-Schnirelmann (Morse concept, Lyusternik-Schnirelmann category)
* 37 Luc Gauthier, Quelques variétés usuelles en géométrie algébrique (algebraic geometry)
* 38 Jean-Louis Koszul, Cohomologie des espaces fibrés différentiables et connexions (Chern-Weil theory)
* 39 Jean Delsarte, Nombre de recommendations des équations polynomiales sur un corps fini, d'après A. Weil (Weil conjectures)
* forty Jacques Dixmier, Anneaux d'opérateurs et représentations des groupes (operator algebras, illustration theory)
* forty-one Roger Godement, Théorie des caractères dans les groupes unimodulaires (unimodular groups)
* forty two Pierre Samuel, Théorie du corps de periods neighborhood selon G. P. Hochschild (local type box theory)
* forty three Laurent Schwartz, Les théorèmes de Whitney sur les fonctions différentiables (singularity theory)
* forty four Jean-Pierre Serre, Groupes d'homotopie (homotopy groups)
* forty five Armand Borel, Cohomologie des espaces homogènes (cohomology of homogeneous areas of Lie groups)
* forty six Samuel Eilenberg, Foncteurs de modules et leurs satellites, d'après Cartan et Eilenberg (homological algebra)
* forty seven Marc Krasner, Généralisations non-abéliennes de los angeles théorie locale des corps de sessions (local fields)
* forty eight Jean Leray, l. a. résolution des problèmes de Cauchy et de Dirichlet au moyen du calcul symbolique et des projections orthogonales et obliques (Dirichlet difficulties and Cauchy difficulties for partial differential equations, symbolic calculus)
* forty nine Pierre Samuel, Sections hyperplanes des variétés normales, d'après A. Seidenberg (algebraic geometry, hyperplane sections, common style)

Additional resources for Differentiable Manifolds: A First Course

Example text

Xr(z)). Here we make the convention that 0 vj (x) = 6 E IRn, even when V j (x) is undefined. Since supp(Xj) c Ujand dom(cpj) = Uj, the expression Xj (x)vj (x) is identically 6 near the set-theoretic boundary of Uj and on all of M \ Uj. This implies that the map i : M + IRk is continuous. Since M is compact and i(M) is Hausdorff, we only need to prove that i is one to one in order to prove that i is a homeomorphism onto its image. Let x, y E M and suppose that i(x) = i(y). Since X is a partition of unity, there is a value of j such that Xj(x) # 0.

If r 2 1, the class Cr (U) of functions f : U -* R that are smooth of order r is specified inductively by requiring that af /axa exist and belong to Cr-I (u), 1 5 i 5 n. The functions that are smooth of order r are also called Crsmooth. One has a decreasing nest and examples show that these inclusions are all proper. 4. The set of infinitely smooth functions on U is It is usual simply to call Cw functions "smooth". We will be concerned primarily with such functions. Remark that the coordinates in U are themselves smooth functions xi : U -, R.

Produce examples to show that both of these inclusions are proper. ) (4) Let U C Wn be open, let f E Cr(U), where 1 5 r 5 oo,and let g : R -* R be Crsmooth also. Prove that the composition g o f belongs to Cr(U). 2. 2. Tangent Vectors We continue to let U C Rn be a fixed but arbitrary open set. We fix p E U and describe the tangent space Tp(U)of U at p. In calculus, it is customary to translate a tangent vector a' at p to the origin 0 E R n , thereby identifying a' canonically with an element of Rn.

Download PDF sample

Rated 4.80 of 5 – based on 45 votes