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Math Help - Submanifold Problem

  1. #1
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    Submanifold Problem

    Let be a differentiable manifold. Show that for each the tangent space is a submanifold of .
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Ricko View Post
    Let be a differentiable manifold. Show that for each the tangent space is a submanifold of .
    Let's see some work! You've posted a lot of questions and have yet to really show your work!

    Which working definition are you using? Immersion by the inclusion map? Plain old that for each x\in T_pM there is a chart (U,\varphi) such that x\overset{\varphi}{\mapsto}\bold{0} and \varphi\left(U\cap T_pM\right)=\left\{x\in\varphi(U):\pi_{k+1}(x)=\cd  ots=\pi_n(x)=0\right\}? That last definition only works for a k-dimensional submanifold of a n-dimensional smooth manifold. Are we working with finite dimensional smooth manifolds?

    There are a lot of questions that need to be answered!
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  3. #3
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    An -dimensional manifold is a submanifold of another -dimensional manifold () if
    a) ( is a subset of )
    b) The inclusion map with is an embedding
    (Which means that for each the differential is and that the inclusion map is a homomorphism)

    So I only care for finite dimensional differential manifolds.

    Well I obviously have (a). Could I use the Reverse function therem for the differential?
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