Abstract
In this chapter, for the first time, we introduce geometry into smooth manifold theory. To define geometric concepts such as lengths and angles on a smooth manifold, we introduce a structure called a Riemannian metric, which is a choice of inner product on each tangent space, varying smoothly from point to point. After defining Riemannian metrics and the main constructions associated with them, we show how submanifolds of Riemannian manifolds inherit induced Riemannian metrics. Then we show how a Riemannian metric leads to a distance function, which allows us to consider connected Riemannian manifolds as metric spaces.
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References
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Lee, J.M. (2013). Riemannian Metrics. In: Introduction to Smooth Manifolds. Graduate Texts in Mathematics, vol 218. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-9982-5_13
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DOI: https://doi.org/10.1007/978-1-4419-9982-5_13
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4419-9981-8
Online ISBN: 978-1-4419-9982-5
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