14.4. Normal Curvature and the Second Fun- damental Form

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Definition 14.4.2 Given a surface X, for any point p = X(u, v) on X, letting L = N · X uu , M = N · X uv , N = N · X vv , where N is the unit normal at p, the quadratic form (x, y) ↦→ Lx 2 +2Mxy+Ny 2 is called the second fundamental form of X at p. It is often denoted as II p . For a curve C on the surface X (parameterized by arc length), the quantity κ N given by the formula κ N = L(u ′ ) 2 + 2Mu ′ v ′ + N(v ′ ) 2 is called the normal curvature of C at p. The second fundamental form was introduced by Gauss in 1827. Normal Curvature . . . Geodesic Curvature . . . Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 694 of 711 Go Back Full Screen Close Unlike the first fundamental form, the second fundamental form is not necessarily positive or definite. Quit
Properties of the surface expressible in terms of the first fundamental are called intrinsic properties of the surface X. Properties of the surface expressible in terms of the second fundamental form are called extrinsic properties of the surface X. They have to do with the way the surface is immersed in E 3 . Normal Curvature . . . Geodesic Curvature . . . Home Page As we shall see later, certain notions that appear to be extrinsic turn out to be intrinsic, such as the geodesic curvature and the Gaussian curvature. Title Page ◭◭ ◮◮ ◭ ◮ This is another testimony to the genius of Gauss (and Bonnet, Christoffel, etc.). Page 695 of 711 Go Back Full Screen Close Quit
Page 1 and 2: 14.4. Normal Curvature and the Seco
Page 3 and 4: The vector −→ t is the unit tan
Page 5 and 6: Thus, it is better to pick a frame
Page 7 and 8: Using the abbreviations X uu = ∂2
Page 9 and 10: Thus, it is clear that the normal c
Page 11: Some authors (including Gauss himse
Page 15 and 16: Furthermore, the principal normal o
Page 17 and 18: We found that the tangential part o
Page 19 and 20: Definition 14.4.4 Given a surface X
Page 21 and 22: However, Normal Curvature . . . κ
Page 23 and 24: Doing so, and remembering that κ
Page 25 and 26: Finally, we get and A = u ′′ 1
Page 27 and 28: Looking at the formulae [α β; γ]
Page 29: From all this, we get 2[α β; γ]

14.4. Normal Curvature and the Second Fun- damental Form

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