Abstract
The most fundamental concept for studying the geometry of \(\mathbb{R}^{2}\) is a straight line. The goal of this chapter is to generalize this fundamental notion from \(\mathbb{R}^{2}\) to arbitrary regular surfaces. Although most surfaces curve in such a way that they don’t contain any straight lines, they do contain curves called geodesics, which will turn out to share many important characterizing properties of straight lines.
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Notes
- 1.
Better-than-normal really is required here; if it were only normal, then γ would be minimizing on [t 0 −ε, t 0] and on [t 0, t 0 +ε], but not necessarily on [t 0 −ε, t 0 +ε]; think about the example S = S 2 with ε = π.
- 2.
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© 2016 Springer International Publishing Switzerland
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Tapp, K. (2016). Geodesics. In: Differential Geometry of Curves and Surfaces. Undergraduate Texts in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-39799-3_5
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DOI: https://doi.org/10.1007/978-3-319-39799-3_5
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Publisher Name: Springer, Cham
Print ISBN: 978-3-319-39798-6
Online ISBN: 978-3-319-39799-3
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