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Elementary Differential Geometry by Christian Bär

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4    The inner geometry of surfaces

We use the notion of an isometry to make the concept of inner geometry of surfaces more precise. Vector fields and their first and second covariant derivatives are introduced. The Theorema Egregrium (‘remarkable theorem’) expresses the Gauss curvature in terms of the curvature tensor and shows the Gauss curvature belongs to the inner geometry of the surface. General Riemann metrics generalise the first fundamental form. The problem of the shortest way from one point to another leads to the concept of the geodesic and the Riemann exponential mapping. In this way it is particularly straightforward to obtain coordinates that are convenient in geometry, like Riemann normal coordinates, geodesic polar coordinates ...

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