Integral Submanifolds
Given a 
-dimensional distribution 
on an 
-dimensional manifold 
, we say that a -dimensional submanifold 
is an “integral submanifold” of if 
for every 
. That is, if the subspace of 
spanned by the images of vectors from 
is exactly 
.
This is a lot like an integral curve, with one slight distinction: in the case on an integral curve we also demand that the length of 
match that of 
, not just the direction (up to sign).
Now, if for every 
there exists an integral submanifold 
of with 
, then is integrable. Indeed, let 
and 
belong to . Since 
is an isomorphism of vector spaces at every point, we can find 
and 
that are 
-related to and , respectively. That is, 
for all 
, and similarly for and . But then we know that ![[X,Y]_{\iota(q)}=\iota_*[\tilde{X},\tilde{Y}]_q](http://s0.wp.com/latex.php?latex=%5BX%2CY%5D_%7B%5Ciota%28q%29%7D%3D%5Ciota_%2A%5B%5Ctilde%7BX%7D%2C%5Ctilde%7BY%7D%5D_q&bg=e6e6e6&fg=333333&s=0 "[X,Y]_{\iota(q)}=\iota_*[\tilde{X},\tilde{Y}]_q")
, and so ![[X,Y]_{\iota(q)}\in\iota_*\mathcal{T}_qN=\Delta_{\iota(q)}](http://s0.wp.com/latex.php?latex=%5BX%2CY%5D_%7B%5Ciota%28q%29%7D%5Cin%5Ciota_%2A%5Cmathcal%7BT%7D_qN%3D%5CDelta_%7B%5Ciota%28q%29%7D&bg=e6e6e6&fg=333333&s=0 "[X,Y]_{\iota(q)}\in\iota_*\mathcal{T}_qN=\Delta_{\iota(q)}")
.
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June 30, 2011 -
Posted by John Armstrong |
Differential Topology, Topology
[...] around such that on . Then the curve with coordinates and for all other is a one-dimensional integral submanifold of through [...]
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[...] a -dimensional distribution on , which we say is “induced” by , and that any connected integral submanifold of should be contained in a leaf of . It makes sense, then, that we should call a leaf of a [...]
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