Abstract
We first introduce “forms” on an n-dimensional vector space V, and then discuss “differential forms” on an n-dimensional manifold M.
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Notes
- 1.
- 2.
A topological space \((X,\mathscr {T})\) is said to be connected if it only has two subsets that are both open and closed (Definition 7 of Sect. 1.2), and is said to be arcwise connected if any two points in X can be joined by a continuous curve in X. A manifold is said to be connected (or arcwise connected) if its base topological space is connected (or arcwise connected). For a topological space, arcwise connected must be connected, but connected is not necessary arcwise connected (there exist “sideswipe” counterexamples). For a manifold, arcwise connected is equivalent to connected [see Abraham and Marsden (1978) Proposition 1.1.33].
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Namely, the Riemann or Lebesgue integral.
- 4.
Integration can also be defined on non-orientable manifolds. In this case, one needs the concept of a “twisted” (also called “odd” or “pseudo”) form, which is outside the scope of this text.
References
Abraham, R. and Marsden, J. (1978), Foundations of Mechanics, Addison-Wesley Publishing Company, Redwood City.
Chern, S. S., Chen, W. and Lam, K. S. (1999), Lectures on Differential Geometry, World Scientific Publishing Company, Singapore.
Wald, R. M. (1984), General Relativity, The University of Chicago Press, Chicago.
Warner, F. W. (1983), Foundations of Differentiable Manifolds and Lie Groups, Springer-Verlag, New York.
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Liang, C., Zhou, B. (2023). Differential Forms and Their Integrals. In: Differential Geometry and General Relativity. Graduate Texts in Physics. Springer, Singapore. https://doi.org/10.1007/978-981-99-0022-0_5
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DOI: https://doi.org/10.1007/978-981-99-0022-0_5
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