By Kapovich M.

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A) To say that the linear dfpo is bijective means that rank(dfpo ) = dim M = dim N (b) The theorem implies that one can find coordinate systems around po and f (po ) so that on the coordinate domain f : M → N becomes the identity map Rn → Rn . The Inverse Function Theorem has two important corollaries which we list themselves as theorems. 12 Immersion Theorem. Let f : M → N be a smooth map of manifolds, n = dim M, m = dim N . Suppose po is a point of M at which the differential dfpo : Tpo M → Tf (po ) N is injective.

In the corresponding linear coordinates (dx1 , · · · , dxn ) on Tp M and (dt1 , · · · , dtm ) on Tp S its differential is given by dx1 = dt1 , · · · , dxm = dtm , dxm+1 = 0, · · · , dxn = 0. This implies the assertion. This lemma may be generalized: . 3 Theorem. Let M be a manifold, f 1 , · · · , f k smooth functions on M . Let S be the set of points p ∈ M satisfying f 1 (p) = 0, · · · , f k (p) = 0. Suppose the differentials df 1 , · · · , df k are linearly independent at every point of S. (a)S is a submanifold of M of dimension m = n − k and its tangent space at p ∈ S is the subspace of vectors v ∈ Tp M satisfying the linear equations (df 1 )p (v) = 0, · · · , (df k )p (v) = 0.

Example. a)Take for f : M → N the radial projection map π : R3 − {0} → S 2 , p → p/ p . If we use the spherical coordinates (θ, φ, ρ) on R3 − {0} and (θ, φ) on S 2 then π becomes the standard projection map (θ, φ, ρ) → (θ, φ) on the coordinate domains . 3. VECTORS AND DIFFERENTIALS 47 b)Take for f : M → N a scalar function f : M → R. This is a submersion at po if dfpo = 0. The submersion theorem asserts that near such a point there are coordinates (xi ) so that f becomes a coordinate function, say f (p) = x1 (p).