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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.
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.
The musical isomorphisms are the global version of this isomorphism and its inverse for the tangent bundle and cotangent bundle of a (pseudo-)Riemannian manifold (,). They are canonical isomorphisms of vector bundles which are at any point p the above isomorphism applied to the tangent space of M at p endowed with the inner product g p ...
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).
243 Quantum Computation and Quantum Information: A Mathematical Perspective, J. M. Landsberg (2024, ISBN 978-1-4704-7557-4) 244 Introduction to Complex Manifolds, John M. Lee (2024, ISBN 978-1-4704-7695-3) 245 Lectures on Differential Geometry, Bennett Chow, Yutze Chow (2024, ISBN 978-1-4704-7767-7)
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A map is a local diffeomorphism if and only if it is a smooth immersion (smooth local embedding) and an open map.. The inverse function theorem implies that a smooth map : is a local diffeomorphism if and only if the derivative: is a linear isomorphism for all points .
In mathematics, and especially gauge theory, Donaldson theory is the study of the topology of smooth 4-manifolds using moduli spaces of anti-self-dual instantons.It was started by Simon Donaldson (1983) who proved Donaldson's theorem restricting the possible quadratic forms on the second cohomology group of a compact simply connected 4-manifold.