Abstract
In this chapter, the basic geometric concepts of Riemannian geometry are introduced. After treating differentiable manifolds, we introduce Riemannian metrics on them. Key concepts are the exponential map, Riemann normal coordinates, and in particular, that of a geodesic. The existence of geodesics is treated with two different methods. The first method is based on the local existence and uniqueness of geodesics. The second method is the heat flow method that gained prominence through Perelman’s solution of the Poincar conjecture by the Ricci flow method.
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Notes
- 1.
Here, we write dF to emphasize the similarity with the derivative of a map between Euclidean domains. In later chapters, we shall rather write DF in order to distinguish the derivative of a map from the exterior derivative of a form.
- 2.
Here, we do not explain that concept, but rather refer to [242] or any advanced textbook in integration theory. For the purposes of this book, it suffices to integrate continuous functions. Likewise, we do not need to consider arbitrary measurable subsets, but only open and closed ones.
- 3.
We shall see a deeper geometric interpretation of the computation leading to (1.6.9) in §9.2B below.
- 4.
These estimates are local estimates on the domain, and therefore, we have to make sure that for suitable regions \(\Omega \times (t_{1},t_{2})\) in S 1 × [0, ∞), the image of u on such a region stays in the same coordinate chart in which we write our Eq. (1.6.4). First of all, since we already have derived a bound on u s , in particular u is uniformly continuous w.r.t. s. As a solution of the heat equation, u is also continuous w.r.t. t so that we may apply the estimates locally in time. The uniform continuity w.r.t. t will be derived shortly.
- 5.
In the bibliography, a superscript will indicate the edition of a monograph. For instance,72017 means 7th edition, 2017.
Bibliography
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Jost, J. (2017). Chapter 1 Riemannian Manifolds. In: Riemannian Geometry and Geometric Analysis. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-319-61860-9_1
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