Abstract
For many problems both in mathematics and beyond, manifolds are the natural class of underlying spaces with which to work. From the perspective of analysis, manifolds are locally indistinguishable from Euclidean spaces, and are therefore tailor-made for use with the tools of analysis. Manifolds provide the natural setting for many concepts from analysis.
We discuss manifolds, submanifolds, smooth maps, tangent spaces and tangent bundles, vector fields and their Lie bracket, and Lie groups. We introduce a number of important examples and include many exercises.
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Notes
- 1.
Real analytic means that the components of f can be locally expressed as convergent power series. The reader who is unfamiliar with real analytic maps may replace real analytic with infinitely differentiable.
- 2.
In this example, x is not a chart, but rather the name of the variables.
- 3.
A term borrowed from Imre Lakatos (1922–1974).
- 4.
Ernst Paul Heinz Prüfer (1896–1934).
- 5.
The Einstein summation convention stipulates that we sum over any indices that occur both as subscripts and as superscripts.
- 6.
Albert Einstein (1879–1955).
- 7.
We denote by \(\mathbb {H}\) the space of Hamilton’s quaternions. We set the convention that vectors in \(\mathbb {H}^n\) are multiplied from the right by scalars in \(\mathbb {H}\), and from the left by matrices in \(\mathbb {H}^{m\times n}\). With this convention, such matrices define \(\mathbb {H}\)-linear maps \(\mathbb {H}^n\to \mathbb {H}^m\). Those who are discomfited by Hamilton’s quaternions may ignore them for the time being.
- 8.
William Rowan Hamilton (1805–1865).
- 9.
Hermann Günther Graßmann (1809–1877).
- 10.
Hassler Whitney (1907–1989).
- 11.
Michel André Kervaire (1927–2007).
- 12.
Marius Sophus Lie (1842–1899).
- 13.
Werner Karl Heisenberg (1901–1976).
- 14.
Our terminology is not uncommon, but there are multiple conventions in the literature.
- 15.
Recall the Einstein summation convention.
- 16.
In the Einstein summation convention, the index i in ∂∕∂x i is counted as a lower index.
- 17.
Carl Gustav Jacob Jacobi (1804–1851).
- 18.
Henri Poincaré (1854–1912), Heinz Hopf (1894–1971).
- 19.
Leonhard Euler (1707–1783).
- 20.
Giuseppe Veronese (1854–1917).
- 21.
Eduard Ludwig Stiefel (1909–1978).
References
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Ballmann, W. (2018). Manifolds. In: Introduction to Geometry and Topology. Compact Textbooks in Mathematics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0983-2_2
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DOI: https://doi.org/10.1007/978-3-0348-0983-2_2
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