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Unformatted text preview: 1.4. VECTOR FIELDS AND FLOWS 23 The integral curves exist only for a small time. If the vector field is not di ff erentiable, then the integral curve is not unique. 1.4.2 Vector Fields on Manifolds Let W be an open subset of a mnifold M and v be a smooth vector field on W . Let ( U , x ) be a local chart in W . If W U , then one can proceed as in R n . If W is not contained in a single chart, then we choose a cover of W and proceed as follows. Let p W and ( U , x ) and ( U , x ) be two charts covering p . Then the integral curves in both local coordinate systems have the same meaning and define a unique integral curve in M . This defines a local flow on W in M . We just need to check that if the flow equations are satisfied in one coordinate system, then they are satisfied in another coordinate system. 1.4.3 Straightening Flows Let U be an open set in a manifold M and t : U M be a local flow on a M such that ( p ) =...
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 Spring '10
 Wong
 Geometry

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