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### Does extending a section by the exponential map make it transverse to the zero set?

Let $V_1, V_2 \rightarrow M$ be two smooth vector bundles over
a smooth riemannian manifold $M$ and $s_1:M\rightarrow V_1$ a section
transverse to the zero set and $s_2: s_1^{-1}(0) \rightarrow V_2 ...

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### General position argument for reasonable spaces

Let $\mathcal{D} \approx \mathbb{P}^{\delta_d}$ be the space of non zero homogeneous degree $d$ polynomials in three variables up to scaling, where
$\delta_d:= \frac{d(d+3)}{2} $. Given a point $p ...

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### What is the easiest way to show that three lines in two dimensional space do not intersect?

I have two similar questions:
1) Let $X$ and $Y$ be two measure spaces. Suppose for every point
$x \in X $ there exists a set $ \mathcal{U}_x \subset Y $ of full
measure in $Y$. Suppose $V \subset ...