Chapter 4 The Intrinsic Geometry of Surfaces

Isometry:

Two surfaces are isometric if there exists a diffeomorphism between these two surfaces whose differential preserves the first fundamental forms of the surfaces. If this can be done only locally, then two surfaces will be called locally isometric. Given two surfaces S₁ and S₂. If we can find parametrizations x₁ and x₂ defined in an open set U of R²for the surfaces S₁ and S₂ respectively such that E₁=E₂, F₁=F₂ and G₁=G₂, then the map x₂x₁^-1 is an isometry from x₁(U) to x₂(U).

Gauss Theorem:

Gaussian curvature K=(eg-f²)/(EG-F²). e, f, g come from the second fundamental form II(v)=-<dN(v), v>. N is the field of normal vectors which describes how the surface S sits in the space R³. Hence a priori the Gaussian curvature depends on the second fundamental form, i.e. on how the surface sits in R³ (it is called extrinsic property). Gauss theorem says that the Gaussian curvature K only depends on E, F, G, i.e. on the first fundamental form (it is called intrinsic property). Hence if two surfaces have the same first fundamental forms, then they will have the same Gaussian curvature. If two surfaces are isometric, then they will have the same Gaussian curvature.

Gaussian curvature K=(eg-f²)/(EG-F²). e, f, g come from the second fundamental form II(v)=-<dN(v), v>. N is the field of normal vectors which describes how the surface S sits in the space R³. Hence a priori the Gaussian curvature depends on the second fundamental form, i.e. on how the surface sits in R³ (it is called extrinsic property). Gauss theorem says that the Gaussian curvature K only depends on E, F, G, i.e. on the first fundamental form (it is called intrinsic property). Hence if two surfaces have the same first fundamental forms, then they will have the same Gaussian curvature. If two surfaces are isometric, then they will have the same Gaussian curvature.

Covariant derivative:

The usual differentiation of a vector field along a curve on a surface may not be a vector field anymore. If we project the derivative of the vector field to the tangent plane at the corresponding point, we get a vector field along the curve, which is called the covariant derivative of the original vector field.

Geodesics:

A curve parametrized by arc length on a regular surface is a geodesic if the covariant derivative of the tangent vector field of the curve is zero. Geodesics are the analogue of straight lines on a curved surface.

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