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Tangent spaces, tangent maps

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Advanced Geometry 2017–2018 ENS de Lyon

Tangent spaces, tangent maps

Exercise 1(Tangent space of a submanifold). Describe the tangent spaceTpM ⊂TpN of the submanifoldM inN at a pointp, for each of the four characterizations of a submanifold.

Exercise 2 (Tangent space of a product). Let M and N be two smooth manifolds, find a natural isomorphism betweenT(p,q)(M×N) and TpM ×TqN.

Exercise 3 (Computation of a differential). Compute the differential ofF :T2→S2 defined as the quotient of the map fromR2 to S2:

F : (x, y)7→(cos(2πx) cos(2πy),cos(2πx) sin(2πy),sin(2πx)).

On which set isF a local diffeomorphism? IsF restricted to this domain a global diffeomor- phism?

Exercise 4. Is it possible to immerse a compact manifoldM of dimensionn >0 inRn?

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