Submanifold Problem

Apr 2010
12
0
Let be a differentiable manifold. Show that for each the tangent space is a submanifold of .
 

Drexel28

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Nov 2009
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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 \(\displaystyle x\in T_pM\) there is a chart \(\displaystyle (U,\varphi)\) such that \(\displaystyle x\overset{\varphi}{\mapsto}\bold{0}\) and \(\displaystyle \varphi\left(U\cap T_pM\right)=\left\{x\in\varphi(U):\pi_{k+1}(x)=\cdots=\pi_n(x)=0\right\}\)? That last definition only works for a \(\displaystyle k\)-dimensional submanifold of a \(\displaystyle n\)-dimensional smooth manifold. Are we working with finite dimensional smooth manifolds?

There are a lot of questions that need to be answered!
 
Apr 2010
12
0
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?