# 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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### Realizing singular homology boundaries as boundary restrictions of manifold maps, in smooth case

Let $X$ be a topological space. Let $\rho = \sum_j \sigma_j$, where the sum is finite, be a chain of singular $k$-simplices in $X$ (in the $k$th singular chain group of $X$, with $\mathbb{Z}$ ...
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### Help proving statement in Lee's Introduction to Riemannian Manifolds about smooth curves into manifolds with nonempty boundary.

The statement appears on page 33 of the second edition of Professor Lee's Introduction to Riemannian Manifolds. It is in the section on Lengths and Distances in Riemannian manifolds, but I think the ...
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### Lee Smooth Manifolds, why does the Whitney Approximation Theorem fail when the co-domain has non-empty boundary?

I am trying to study chapter 6 of Lee's Introduction to Smooth Manifolds. In a remark after the Whitney Approximation Theorem, Lee stated that this theorem do not hold because it might not be possible ...
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2 votes
2 answers
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### General condition for the surface element be the same as the volume element, up to a dt

The surface element in spherical coordinates is $r \sin \theta \mathrm{d}\theta \mathrm{d}\varphi$, and the volume element is $r \sin \theta \mathrm{d}\theta \mathrm{d}\varphi \mathrm{d}r$. We see ...
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1 answer
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### Understanding difference between a distinguished boundary and 'normal' boundary in several complex variables

I am reading through Tasty Bits of Several Complex Variables and I come across the term distinguished boundary. It seems a distinguished boundary is different from a normal boundary as the author ...
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2 votes
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### Normal derivative and Hodge star operator

Let $(M,g)$ be an oriented Riemannian manifold with boundary $\partial M$. Let $j\colon \partial M\to M$ denotes the natural embedding and $\mu_M$ the canonical volume form. If $\psi\in C^\infty(M)$ ...
1 vote
1 answer
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### Does the 3-manifold $S^1\times S^2$ bound a smooth integral homology ball?

Does the 3-manifold $S^1\times S^2$ bound a smooth (integral) homology ball? The only 4-manifolds I know whose boundary is $S^1\times S^2$ are $S^1\times D^3$ and $D^2\times S^2$, and both are not ...
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1 answer
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### Integration by parts on compact, non-orientable Riemannian manifold with boundary

Let $(M,g)$ be a compact Riemannian manifold, not necessarily orientable or without boundary. Let $\mu$ be a normalized volume measure on $M$ and $u$ be a smooth function on $M$. In some notes that I ...
1 vote
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### Flat Riemannian manifolds with boundary

The universal cover of a flat, connected, complete Riemannian manifold is Euclidean space, which allows for classification of all such manifolds as quotient spaces $\mathbb{R}^n/\Gamma$ for ...
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### Preimage of a regular value by smooth function on bounded manifold

Suppose $f:M\to\mathbb{R}$ is a smooth function, where $M$ is a smooth, bounded manifold with boundary, $\partial M$. Suppose we knew $f_{|\partial M}=0$ and that some $c\in\mathbb{R}-\{0\}$ is a ...
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1 vote
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### Stokes' theorem for a manifold without boundary

Stokes' theorem states that when $M$ is a compact oriented $m$-manifold with boundary, and $\omega$ is a $(m-1)$-form on $M$, we have $$\int_{\partial M}\omega = \int_{M} d\omega.$$ This is confusing ...
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1 vote
1 answer
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### Heat kernel expansion for space with boundaries but with no particular boundary conditions

In the document Heat kernel expansion, user's manual Vassilevich expresses in equations $(5.29-5.33)$ the coefficients of the heat kernel expansion up to the fourth one for a manifold with boundaries. ...
2 votes
1 answer
99 views

### Top de Rham cohomology group for noncompact manifolds with boundary

Suppose that $M$ is a smooth, connected, oriented $m$-manifold with (empty or nonempty) boundary. I am aware that top de Rham cohomology group $H^m_{\mathrm{dR}}(M;\mathbb{R})$ is trivial for ...
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### The Submersion Level Set Theorem for Manifolds with Boundary

We have the following theorem. (Lee, Introduction to Smooth Manifolds). Theorem. Let $M$ and $N$ be smooth manifolds. Let $F : M \to N$ be a submersion. Then each level set of $F$ (i.e., $F^{-1}(c)$) ...
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1 vote
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### Apparent contradiction with gluing 2-handle to a 4-manifold all being isotopic.

In Hirsch's Differential Topology Chapter 8 Theorem 2.3, it says : Let $f, g:\partial Q\approx \partial P$ be isotopic diffeomorphisms. Then $P\cup_f Q\approx P\cup_g Q$. Here $\approx$ means ...
1 vote
1 answer
56 views

### Differentiable structure on the boundary induces differentiable structure on the entire manifold

I am trying to prove the uniqueness of smoothed corners, in detail. In Milnor's notes, https://www.maths.ed.ac.uk/~v1ranick/papers/homsph.pdf, there's an explanation in p.34-37 on how to do it. The ...
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### Smooth maps on manifold with nonconstant push forward maps

I am a beginner in manifold. I am reading Push forward map ( or differential map at a point). I came across the following question: Give smooth maps $F: M \to \mathbb{R^3}$ and $G: N \to \mathbb{R^3}$ ...
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### Tangent space at a boundary point of closed disc.

I am a beginner at Manifolds. I am reading Tangent space at a point of a manifold $M$. What I know: If I take my manifold $M$ to be a closed unit disk then for any point $a$ inside the disk I know ...
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1 vote
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### Relative homology of $M\times M$ to diagonal

Let $M$ be a compact manifold. I want to understand why $$\beta_i(M\times M, M) = \beta_i(M\times M)-\beta_i(M),$$ where $\beta_i$ denotes the $i$th Betti number and we identify $M$ with the ...
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### Kirby Diagram for the Trefoil

I'm studying R.E. Gompf and A.I. Stipsticz, 4-Manifolds and Kirby Calculus and I got stuck with a question. Let $K$ be the right-handed trefoil embedded in $\partial \mathbb{D}^4$, we know that, ...
1 vote
1 answer
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### Question on Bredon's Theorem 11.9 about the duality of intersection product and cup product

Let $i_K^W:K^k\to W^w$, $i_N^W:N^n\to W^w$ be smooth embeddings of smooth manifolds with boundary and assume that $N$ resp. $K$ meet $\partial W$ transversely in $\partial N$ resp. $\partial K$. 11.8 ...
1 vote
1 answer
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### If $M$ is smooth $k$-manifold with corners then is $M$ without corner points a smooth $k$-manifold with boundary?

First of all we remember some elementary definitions and results about manifolds with corners. Definition A function $f$ defined in a subset $S$ of $\Bbb R^k$ is said of class $C^r$ if it can be ...
2 votes
1 answer
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### Is the boundary of a $k$-manifold with corners a $(k-1)$-manifold with corners too?

First of all we remember some elementary definitions and results about manifolds with corners. Definition A function $f$ defined in a subset $S$ of $\Bbb R^k$ is said of class $C^r$ if it can be ...
0 votes
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### The notation for the boundary of a region

For a region $A$ in $n$space, we use the notation $\partial A$ to denote the boundary of the region $A$, but why we are using the symbol $\partial$, is there some connection between the partial ...
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2 votes
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