7
votes
Transversality and $C^l$, $C^{\infty}$ spaces of almost complex structures
There are several different issues happening here. The space of $C^\infty$ is not a Banach space because the topology is generated by a countable family of semi-norms not by one norm. You might try to ...
- 3,429
6
votes
Accepted
perturbing one map to be transverse to a second map
I'm not sure what exactly you mean by perturb, but you can always make $f$ transverse to $g$ by a homotopy and this is often enough. This is proved in Section IV.2 of Kosinski's "Differential ...
- 35.9k
4
votes
Accepted
Almost geodesic on non complete manifolds
Start with the plane $\mathbb R^2$ and remove a slab, but keep a line going through the slab:
$$ Slab = \{(x, y) \in \mathbb R^2 : 0 < y < 1, x \neq 0\} $$
$$ M = \mathbb R^2 - Slab$$
...
- 856
3
votes
Accepted
Transversal pre-image of a small enough trivial tubular neighborhood contains a trivial tubular neighborhood
The condition $\mathcal P(\epsilon)$ is needlessly complicated. It's always true without passing to components.
Let $U$ be a tubular neighborhood of $S$ diffeo to $[-1,1]\times S$. Let $\pi: U\to \...
- 7,842
2
votes
Accepted
Finite-dimensional argument for Morse-Smale pairs?
The Sard-Smale result certainly guarantees that this will be true for a generic 𝑔, but will it hold for any 𝑔?
If $g$ is fixed, you can certainly use the Sard-Smale theorem to prove the existence ...
1
vote
Every immersion can be deformed to have only transverse self-intersections
The statement follows easily from Sard's lemma.
Suppose $k=\dim N-\dim N$.
Cover $M$ by a countable collection of charts $s_i\:U_i\to M$ such that $f\circ s_i$ extends to embeddings $U_i\times \mathbb{...
- 45k
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