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I am reading the unit on manifolds in Flemings Functions of Several Variables. Here's a snippet from the page:

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I am not sure of the last paragraph: how is the dimension of the intersections of the tangent spaces $r + s - n$. Also, shouldn't it be the case that the intersection of the tangent spaces is empty. If $\mathbf{h}$ belongs to the intersection,

$\mathbf{h} . \text{grad} \mathbf{\Psi} = 0$ and $\mathbf{h} . \text{grad} \mathbf{\Theta} = 0$,

and this will make the gradients to be linearly dependent, which shouldn't be the case.

Junaid Aftab
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  • What does $h. \text{grad} \Psi$ mean ? – Max Apr 05 '17 at 15:15
  • A vector $\mathbf{h}$ dot product-ed with the gradient of the first vector valued function $\mathbf{\Psi}$, at a point that lies in the intersection of the manifolds. – Junaid Aftab Apr 05 '17 at 15:18
  • Hmm. Do the dimensions really match? It does't seem like you can define $h. \text{grag} \Psi$ or $h.\text{grad}\Theta$. – Max Apr 05 '17 at 15:20
  • Of the individual tangent spaces? No they don't, since both the manifolds have different dimensions. But I'm still or sure how is the dimension of the intersection of the spaces defined and why if a vector is in the intersection of the tangent spaces, wouldn't it case any issue? – Junaid Aftab Apr 05 '17 at 15:24
  • I'm not sure if this is too basic but: We are picking functions $\Phi$ which realize $M$ as a zero set, and similarly for $\Psi$ and $N$. Then when both of the vector functions vanish, we are in $M \cap N$. Assuming $M$ and $N$ intersect transversely gives the right dimension for the intersection of tangent spaces. – Max Apr 05 '17 at 15:33
  • I get that, minus the transversely bit; I haven't done topological manifolds as of yet so I don't really understand what it means. I'm still confused though: why is the dimension of the intersection of the tangent spaces equal to $r + s - n$. I get that is consistently reproduces the correct dimension of the manifolds, and the relation is pinned by the rank nullity thoerem. But the we also have using dim($X_1 + X_2$) + dim ($X_1 \ cap X_2$) = dim($ X_1 $) + dim($X_2$). If the definition is true, that will imply that dimension of the sum of tangent spaces is $n$. How did this follow – Junaid Aftab Apr 05 '17 at 15:40
  • Assuming that the dimension of the intersection is $r+s-n$ is equivalent to assuming the sum of the tangent spaces has dimension $n$ (because of the equality you state). You have to assume one of those conditions or bad things can happen. Either condition is the same thing as "transversality." – Max Apr 05 '17 at 15:51
  • Yes, you make that assumption beforehand- it is not guaranteed. Consider the intersection of (the graphs of) $y=x^2$ with $y=0$. In that case $\Theta$ is not full rank and the tangent spaces coincide. – Max Apr 05 '17 at 15:54

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