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differential geometry w notes from teacher_Part_2

# differential geometry w notes from teacher_Part_2 - 3 1.1...

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1.1. SUBMANIFOLDS OF EUCLIDEAN SPACE 3 be a subset of the Euclidean space R m described by the locus of r equations F α ( x 1 , . . . , x m ) = y α 0 , α = 1 , . . . , r . Suppose that M is non-empty and let x 0 M , that is F ( x 0 ) = y 0 . Then the Implicit Function Theorem says that if det F α x β ( x 0 ) 0 , where α = 1 , . . . , r and β = n + 1 , . . . , n + r , then there is a neighborhood of x 0 such that the last r coordinates can be expressed as smooth functions of the first n coordinates: x α = f α ( x 1 , . . . , x n ) , α = n + 1 , . . . , n + r . If this is true for any point of M , then M is a n -dimensional submanifold of R m . The matrix F α x j , where α = 1 , . . . , r and j = 1 , . . . , m is called the Jacobian matrix . The General Implicit Function Theorem says that if at a point x 0 the Ja- cobian matrix has the maximal rank equal to r , rank F α x j ( x 0 ) = r , then there exists a coordinate system in a neighborhood of x 0 such that the last r coordinates can be expressed as smooth functions of the first n coor- dinates. If this is true for every point of M , then M is a n -dimensional submanifold of R m . The number r is called the codimension of M . If the codimension r is equal to 1, that is n = m - 1, then M is called a hypersurface . di ff geom.tex; April 12, 2006; 17:59; p. 7

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