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Math Help - question on pushforward on tangent vectors

  1. #1
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    question on pushforward on tangent vectors

    in my book tangent vectors are defined as equivalence classes of curves that are tangent to the given vector. let σ be a curve on a manifold and h*([σ]) is defined to be [h o σ]. v1 = [σ1] and v2 = [σ2]. show that the pushforward is linear: h*(v1 + v2) = h*(v1) + h*(v2) and h*(rv1) = rh*(v1).

    i am starting this problem with showing the first part h*(v1 + v2) = h*(v1) + h*(v2) and i am having trouble. earlier my book defines v1 + v2 = [φ^-1 (φ o σ1 + φ o σ2)] where φ is a coordinate function.

    i have that h*(v1+v2) = [h(φ^-1 (φ o σ1 + φ o σ2))] but i don't know how to get from there to h*(v1) + h*(v2) = [h o σ1] + [h o σ2]. if φ^-1 was linear then it would work but that doesn't seem to be the case all the time however.
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    MHF Contributor Bruno J.'s Avatar
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    Re: question on pushforward on tangent vectors

    Hey oblixps. I'll gladly help you with your question if you'll be so kind so as to reword it in LaTeX so that I can read it without getting a major headache.

    Thanks!
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