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02 - tangent vectors

02 - tangent vectors - INTRODUCTION TO MANIFOLDS II Tangent...

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INTRODUCTION TO MANIFOLDS — II Tangent Bundles 1. Tangent vectors, tangent space. Let M n be a smooth n -dimensional manifold, endowed with an atlas of charts x : U R n , y : V R n , . . . , where M = U V ∪ · · · are domains of the corre- sponding charts. Definition. Two smooth curves ϕ i : ( - ε, ε ) M , i = 1 , 2, passing through the same point p M , are said to be 1-equivalent , ϕ 1 ϕ 2 , if in some chart x : U R n k x ( ϕ 1 ( t )) - x ( ϕ 2 ( t )) k = o ( t ) as t 0 + . (1) Problem 1. Prove that the condition (1) is actually independent of the choice of the chart. Definition. The tangent space to the manifold M at the point p is the quo- tient space C 1 ( R 1 , M ) / by the equivalence (1). Notations : the equivalence class of a curve ϕ will be denoted by [ ϕ ] p . Instead of saying that a curve ϕ belongs to a certain equivalence class v = [ · ] p , we say that the curve ϕ is tangent to the vector v . Problem 2. Prove that the tangent space at each point is isomorphic to the arithmetic space R n . Solution. Fix any chart x around the point p and consider the maps f iso , iso defined as f iso: R n C ( R , M ) , v = ( v 1 , . . . , v n ) 7→ ϕ v ( · ) , ϕ v ( t ) = x - 1 ( x ( p ) + tv ) (2) iso( v ) = [ f iso( v )] p . (3) This map is injective (prove!). To prove its surjectivity, for any smooth curve ϕ consider its x -coordinate representation, x ( ϕ ( t )) = x ( p ) + tv + · · · , existing by virtue of differentiability of the latter. Then f iso( v ) ϕ . Remark. This is a good example of abstract nonsense! The idea is that you associate with each curve its linear terms, the coordinate system being fixed. Then any curve is uniquely defined by its linear terms up to the 1-equivalence, since the definition (1) was designed especially for this purpose! Typeset by A M S -T E X 1
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2 TANGENT BUNDLES Definition. The string of real numbers ( v 1 , . . . , v n ) is called the coordinate representation of the tangent vector [ ϕ ] p in the coordinate system x . Problem 3. If M n is a hypersurface in R n +1 , then the tangent space is well de- fined by geometric means. Prove that this “geometric” tangent space is isomorphic to the one defined by the abstract definition above. A good exercise for practicing in abstract nonsense! Important note: The coordinate system x occurs in the construction of iso- morphisms (2), (3) in the most essential way! If another coordinate system is chosen, then the isomorphisms will be completely different.
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