By Loring W. Tu
Manifolds, the higher-dimensional analogs of gentle curves and surfaces, are primary items in glossy arithmetic. Combining elements of algebra, topology, and research, manifolds have additionally been utilized to classical mechanics, normal relativity, and quantum box theory.
In this streamlined advent to the topic, the speculation of manifolds is gifted with the purpose of assisting the reader in achieving a swift mastery of the fundamental themes. via the tip of the publication the reader can be in a position to compute, at the least for easy areas, essentially the most simple topological invariants of a manifold, its de Rham cohomology. alongside the way in which the reader acquires the information and abilities priceless for extra research of geometry and topology. The considered necessary point-set topology is integrated in an appendix of twenty pages; different appendices overview evidence from genuine research and linear algebra. tricks and strategies are supplied to some of the workouts and problems.
This paintings can be utilized because the textual content for a one-semester graduate or complex undergraduate direction, in addition to via scholars engaged in self-study. Requiring simply minimum undergraduate prerequisites, An Introduction to Manifolds can be an outstanding beginning for Springer GTM eighty two, Differential kinds in Algebraic Topology.
Read or Download An Introduction to Manifolds PDF
Similar differential geometry books
Within the Spring of 1966, I gave a sequence of lectures within the Princeton college division of Physics, aimed toward contemporary mathematical leads to mechanics, specially the paintings of Kolmogorov, Arnold, and Moser and its program to Laplace's query of balance of the sun process. Mr. Marsden's notes of the lectures, with a few revision and growth by way of either one of us, turned this ebook.
Very good short advent offers primary idea of curves and surfaces and applies them to a few examples. issues comprise curves, idea of surfaces, primary equations, geometry on a floor, envelopes, conformal mapping, minimum surfaces, extra. Well-illustrated, with ample difficulties and options.
This quantity includes thirty-six examine articles provided at the Colloquium on Differential Geometry, which was once held in Debrecen, Hungary, July 26-30, 1994. The convention used to be a continuation in the sequence of the Colloquia of the J? nos Bolyai Society. the variety lined displays present job in differential geometry.
In the course of its first hundred years, Riemannian geometry loved regular, yet undistinguished development as a box of arithmetic. within the final fifty years of the 20th century, besides the fact that, it has exploded with task. Berger marks the beginning of this era with Rauch's pioneering paper of 1951, which incorporates the 1st actual pinching theorem and an grand jump within the intensity of the relationship among geometry and topology.
- Lectures on minimal surfaces. Vol.1, Introduction, fundamentals, geometry
- Integral geometry, Radon transforms and complex analysis: Lectures
- Principal Bundles: The Classical Case (Universitext)
- The Ricci Flow: An Introduction (Mathematical Surveys and Monographs)
- Vector bundles
Extra info for An Introduction to Manifolds
Vσ (k) and leave the arguments of g alone; for all τ in Sk , the resulting permutations σ τ in Sk+ contribute the same term to the sum since (sgn σ τ )f (vσ τ (1) , . . , vσ τ (k) ) = (sgn σ τ )(sgn τ )f (vσ (1) , . . , vσ (k) ) = (sgn σ )f (vσ (1) , . . , vσ (k) ), where the first equality follows from the fact that (τ (1), . . , τ (k)) is a permutation of (1, . . , k). So we divide by k! to get rid of the k! repeating terms in the sum coming from the permutations of the k arguments of f ; similarly, we divide by !
9. Commutator of derivations and antiderivations k Let A = ⊕∞ k=0 A be a graded algebra over a field K. A superderivation of A of degree m is a K-linear map D : A − → A such that D(Ak ) ⊂ (Ak+m ) and for all a ∈ Ak and b∈A , D(ab) = (Da)b + (−1)km a(Db). If D1 and D2 are two superderivations of A of respective degrees m1 and m2 , define their commutator to be [D1 , D2 ] = D1 ◦ D2 − (−1)m1 m2 D2 ◦ D1 . Show that [D1 , D2 ] is a superderivation of degree m1 + m2 . (A superderivation is said to be even or odd depending on the parity of its degree.
In summary, on an open subset U of R3 , there are identifications 0 (U ) ⏐ ⏐ d −−−−→ 1 (U ) ⏐ ⏐ d −−−−→ 2 (U ) ⏐ ⏐ d −−−−→ 3 (U ) ⏐ ⏐ C ∞ (U ) −−−−→ X(U ) −−−−→ X(U ) −−−−→ C ∞ (U ). grad curl div Propositions A and B express the property d 2 = 0 of the exterior derivative. A vector field P , Q, R on R3 is the gradient of a C ∞ function f if and only if the corresponding 1-form P dx + Q dy + R dz is df . Proposition C expresses the fact that a 1-form on R3 is exact if and only if it is closed. On the other hand, Proposition C need not be true on a region other than R3 , as the following well-known example from calculus shows.