I'm reading about the Gauss-Bonnet theorem, and in the beginning of the section I'm studying I saw:
Why is this true? I know this should be trivial but I'm not seeing it.
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Matheus Andrade
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The cited paragraphs make very little sense. What book are you reading? – Moishe Kohan Jul 06 '19 at 06:36
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@MoisheKohan: the book is "A first course in curves and surfaces", from Ted Shifrin. Page 79 here http://alpha.math.uga.edu/~shifrin/ShifrinDiffGeo.pdf – Matheus Andrade Jul 06 '19 at 17:44
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4In fairness, I was rather misquoted. – Ted Shifrin Jul 06 '19 at 19:14
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@TedShifrin sorry, I fixed it now. – Matheus Andrade Jul 06 '19 at 19:28
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Now the claim is clearly true and you should try to write a proof yourself starting with formalization of the notion "turns relative to the given framing". – Moishe Kohan Jul 06 '19 at 19:55
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@MoisheKohan The angle between parallel vector fields is constant and if $\alpha$ is a geodesic then $\alpha'(t)$ is a parallel vector field. So if $v$ is tangent to $\alpha$, $\langle \alpha'(t), v \rangle$ is constant. Is that it? – Matheus Andrade Jul 06 '19 at 20:05
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Yes, because it is zero. – Moishe Kohan Jul 07 '19 at 00:28