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Elementary Differential Geometry by Christian Bär

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3    Classical surface theory

We introduce regular surfaces and their tangent planes. We investigate what normal fields have to do with orientability of surfaces. We discover that the geometry of regular surfaces is largely determined by the first and second fundamental forms, which also give rise to different types of curvature. We learn how to integrate over regular surfaces and, in particular, how their surface area is defined. Finally we examine some special classes of surfaces more closely: ruled surfaces, minimal surfaces, surfaces of revolution and tubular surfaces.

3.1    Regular surfaces

Surfaces in three-dimensional space are two-dimensional objects, i.e. the points on a surface can be described by two independent parameters. The definition ...

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