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Introduction to Topological Manifolds, Springer-Verlag, Graduate Texts in Mathematics 2000, 2nd edition 2011 [5] Lee, John M. (2012). Introduction to Smooth Manifolds. Graduate Texts in Mathematics. Vol. 218 (Second ed.). New York London: Springer-Verlag. ISBN 978-1-4419-9981-8. OCLC 808682771.
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Lee, John M., Introduction to Smooth Manifolds, Springer-Verlag, New York (2003) ISBN 0-387-95495-3.Graduate-level textbook on smooth manifolds. Hwa-Chung, Lee, "The Universal Integral Invariants of Hamiltonian Systems and Application to the Theory of Canonical Transformations", Proceedings of the Royal Society of Edinburgh.
Michele Audin, Torus actions on symplectic manifolds, Birkhauser, 2004 John Lee, Introduction to smooth manifolds , chapter 9, ISBN 978-1-4419-9981-8 Frank Warner, Foundations of differentiable manifolds and Lie groups , chapter 3, ISBN 978-0-387-90894-6
Lee, John M. (2003). Introduction to smooth manifolds. New York: Springer. ISBN 0-387-95448-1. A textbook on manifold theory. See also the same author's textbooks on topological manifolds (a lower level of structure) and Riemannian geometry (a higher level of structure).
In mathematics, particularly topology, an atlas is a concept used to describe a manifold. An atlas consists of individual charts that, roughly speaking, describe individual regions of the manifold. In general, the notion of atlas underlies the formal definition of a manifold and related structures such as vector bundles and other fiber bundles.
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It is again a smooth vector field. The metric g can take up to two vectors or vector fields X, Y as arguments. In the former case the output is a number, the (pseudo-)inner product of X and Y. In the latter case, the inner product of X p, Y p is taken at all points p on the manifold so that g(X, Y) defines a smooth function on M. Vector fields ...