EDG Ch.6 -- Geometry of Surfaces in

Introduction to Chapter 6: Geometry of Surfaces in R3

Now that we know how to measure the shape of a surface M in R3, the next step is to see how the shape of M is related to its other properties. Near each pointof M, the Gaussian curvature has a strong influence on shape (Rmk. 3.3 of Ch.5), but we are now interested in the situation in the large--over the whole extent of M. For example, what can be said about the shape of M if it is compact, or flat, or both?

In the early 1800s Gauss raised a question that led to a new and deeper understanding of what geometry is: How much of the geometry of a surface in R3 is independent of its shape? At first sight this seems a strange question--what can we possibly say about a sphere, for example, if we ignore the fact that it is round? To get some grip on Gauss's question, let us imagine that the surface M in R3 has inhabitants who are unaware of the space outside their surface and thus have no conception of its shape in R3.

Nevertheless, they will still be able to measure the distance from place to place in M and find the area of regions in M. We shall see that in fact they can construct an intrinsic geometry for M that is richer and no less interesting than the familiar geometry of the plane R2.

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