# Questions tagged [manifolds-with-boundary]

For questions about manifolds with boundaries, as well as manifolds with corners, and other such generalisations of the notion of a manifold.

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### What is the boundary of a tubular neighbourhood of the projective plane embedded in $\mathbb{R}^4$?

There are various ways to embed the projective plane into $\mathbb{R}^4$ very nicely, see e.g. Wikipedia. Suppose now that I take such an embedded projective plane $P \subset \mathbb{R}^4$ and fix a ...
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### How can we apply generalized Stokes' theorem to a non-oriented manifold with boundary?

I do not really know much about the boundary of non-oriented manifold. A boundary of oriented manifold, if it exists, has a sign. If you reverse the orientation, the boundary picks up an extra ...
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### geodesic balls in Riemannian manifolds with bounded geometry

Let $(M,g)$ be an open (:=complete, non-compact) Riemannian manifold with bounded geometry, in the sense that in some atlas of charts of radius $r_0>0$, the metric and all its derivatives are ...
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### Searching for a connected neighborhood of fixed radius for every point of a submanifold

Let $\Gamma=\gamma(\partial D)\subset \mathbb R^3$ be an image of $C^1$-curve, where $D$ is an open ball in $\mathbb R^2$. My question is whether it is possible to find a fixed $\epsilon >0$ such ...
### Spivak manifolds - definition of $dw$ for a p-form $w$ on a manifold $M$
Spivak says the definition of $dw$ for a k-form $w$ does not make sense on a manifold because $D_j(w_{i_1, \dots , i_p})$ has no meaning. Does it have no meaning because the function w_{i_1, \dots , ...