differential geometry w notes from teacher_Part_65

differential geometry w notes from teacher_Part_65 - 129...

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5.4. DEGREE OF A MAP 129 Then N is a vector field that is everywhere normal to M . Notice that the norm of the normal vector is || N || 2 = || e 1 || 2 || e 2 || 2 - ( e 1 , e 2 ) 2 . The second fundamental form , or the extrinsic curvature is defined by the matrix b μν = 1 || N || e μ u ν , N = 1 || N || 2 x i u μ u ν N i . The second fundamental form describes the extrinsic geometry of the sur- face M . The mean curvature of M is defined by H = g μν b μν . The Gauss curvature of M is defined by K = det b μν det g αβ . Gauss has shown that the K is an intrinsic invariant. In fact, K = R 12 12 = 1 2 R , where R 12 12 is the only non-vanishing components of the Riemann curvature of the metric g and R is the scalar curvature. This will be discussed later. The Gauss map is the map ϕ : M S 2 from M to S 2 defined by ϕ ( x ) = N ( x ) || N ( x ) || , that is, it associates to every point x in M the unit normal vector at that point.
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