Analysis and Algebra on Differentiable Manifolds: A Workbook for Students and Teachers
Springer Science & Business Media, Dec 12, 2009 - Mathematics - 478 pages
A famous Swiss professor gave a student’s course in Basel on Riemann surfaces. After a couple of lectures, a student asked him, “Professor, you have as yet not given an exact de nition of a Riemann surface.” The professor answered, “With Riemann surfaces, the main thing is to UNDERSTAND them, not to de ne them.” The student’s objection was reasonable. From a formal viewpoint, it is of course necessary to start as soon as possible with strict de nitions, but the professor’s - swer also has a substantial background. The pure de nition of a Riemann surface— as a complex 1-dimensional complex analytic manifold—contributes little to a true understanding. It takes a long time to really be familiar with what a Riemann s- face is. This example is typical for the objects of global analysis—manifolds with str- tures. There are complex concrete de nitions but these do not automatically explain what they really are, what we can do with them, which operations they really admit, how rigid they are. Hence, there arises the natural question—how to attain a deeper understanding? One well-known way to gain an understanding is through underpinning the d- nitions, theorems and constructions with hierarchies of examples, counterexamples and exercises. Their choice, construction and logical order is for any teacher in global analysis an interesting, important and fun creating task.
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atlas automorphism basis canonical change of coordinates chart complex manifold components Compute connection form Consider constant curvature coordinate system cosh curvature form curvature tensor defined Definitions denotes diffeomorphism differentiable manifold differentiable structure differential form element embedding Euclidean metric fibre formula functions geodesic given GL(n Hence holomorphic homeomorphism identity immersion injective integral curves isometry isomorphism Jacobian left-invariant Let G Levi-Civita connection Lie algebra Lie group linear connection linear frames linear group map f matrix metric g Moreover n-manifold neighborhood obtain open subset orientable orthogonal orthonormal parametrization principal bundle Problem Prove quotient manifold resp respect Riemannian manifold sectional curvature SL(n SO(n Solution Sp(n submanifold submersion tangent space tensor field Theorem topology trivial vector bundle vector field vector space