By Wulf Rossmann

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Every p ∈ S lies in some domain U S, by definition. 52 CHAPTER 1. MANIFOLDS Let S be a submanifold of M , and i : S → M the inclusion map. The differential dip : Tp S → Tp M maps the tangent vector of a curve p(t) in S into the tangent vector of the same curve p(t) = i(p(t)), considered as curve in M . We shall use the following lemma to identify Tp S with a subspace of Tp M . 2 Lemma. Let S be a submanifold of M . Suppose S is given by the equations xm+1 = 0, · · · , xn = 0 ⊂ locally around p.

E. p = (0, 0, 0). Hence the differential d(x2 + y 2 − z 2 ) = 2(xdx + ydy − zdz) is everywhere non−zero on S − {(0, 0, 0)} which is therefore is a (3 − 1)-dimensional submanifold. Proof of (b) (by contradiction). ) S were a 2-dimensional submanifold of R3 . Then the tangent vectors at (0, 0, 0) of curves in R3 which lie on S would form a 2-dimensional subspace of R3 . But this is not the case: for example, we can find three curves of the form p(t) = (ta, tb, tc) on S whose tangent vectors at t = 0 are linearly independent.

6 Theorem. Let M be an n-dimensional manifold, g : Rm → M , (t1 , · · · , tm ) → g(t1 , · · · , tm ), a smooth map. Suppose the differential dguo has rank m at some point uo .