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Unformatted text preview: 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 nonvanishing 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...
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This note was uploaded on 11/26/2011 for the course MAT 4821 taught by Professor Wong during the Spring '10 term at FSU.
 Spring '10
 Wong
 Geometry

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