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differential geometry w notes from teacher_Part_3

# differential geometry w notes from teacher_Part_3 - 5 1.1...

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1.1. SUBMANIFOLDS OF EUCLIDEAN SPACE 5 1.1.3 Main Theorem on Submanifolds of R m The matrix of the linear transformation F * is exactly the Jacobian matrix. Therefore, the di ff erential F * at a point x 0 is a surjective (onto) map if and only if m r and the Jacobian at x 0 has the maximal rank equal to r . Recall that F * : R m x 0 R r y 0 is surjective if for any w R r y 0 there is v R m x 0 such that F * v = w . Thus, we have the following theorem. Theorem 1.1.1 Let F : R m R r with m > r, y 0 R r and M = F - 1 ( y 0 ) = { x R m | F ( x ) = y 0 } . If M is non-empty and for any x 0 M the di ff erential F * : R m x 0 R r y 0 is surjective, then M is a n = ( m - r ) -dimensional submanifold of R m . di ff geom.tex; April 12, 2006; 17:59; p. 9

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6 CHAPTER 1. MANIFOLDS 1.2 Manifolds 1.2.1 Basic Notions of Topology First we define the basic topological notions in the Euclidean space R n . Let x 0 R n be a point in R n and ε > 0 be a positive real number.
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