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 and Fermi coordinates. Jacobi fields illustrate the inner geometric importance of the Gauss curvature. Spherical and hyperbolic geometry are investigated in more detail. Their trigonometry is derived and applications to cartography are discussed. The hyperbolic plane satisfies all axioms of Euclidean geometry except for the parallel axiom.
When we consider surfaces in ℝ3 we tend to pay special attention to their relative geometries, i.e. to how the surface is embedded into the surrounding space. We quasi look at them from outside. One could also try to imagine oneself in the position of a (two-dimensional) inhabitant of the surface, and examine those properties of the surface that can be observed by a being who cannot peek out of the surface.