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I'm looking for a quick introduction to the idea of an angle function for a manifold. As a guess I think it's similar to a distance function, but maps tangent vectors rather than points to some value.

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... you must know something about inner products, if you know about manifolds? Differentiable manifolds in general are not equipped with either a notion of distance or angle. For a notion of angle you need at least a conformal structure. –  Zhen Lin Jul 23 '11 at 14:12
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Related: math.stackexchange.com/questions/53291/… / Each tangent space on a Riemannian manifold is isomorphic to a Euclidean space with canonical metric. The Riemannian structure is sufficient, but not necessary, to define angles. See Zhen's comment. –  Willie Wong Jul 23 '11 at 14:16
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As Zhen says, a topological or smooth manifold is not equipped with a notion of angle. To describe an angle requires a conformal structure on a manifold, which can be induced by a Riemannian metric. A Riemannian metric equips tangent spaces with an inner product, so one can speak of the angle between two tangent vectors.

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