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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.

Manifolds are typically defined to be without boundaries (every point has a neighbourhood homeomorphic to an Euclidean open disc), and this tag is for questions about manifolds with boundaries, as well as manifolds with corners, and other such generalisations of the notion of a manifold.

618 questions
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Why are functions with vanishing normal derivative dense in smooth functions?

Question Let $M$ be a compact Riemannian manifold with piecewise smooth boundary. Why are smooth functions with vanishing normal derivative dense in $C^\infty(M)$ in the $H^1$ norm? Here I define $C^\infty(M)$ to be those functions which have all…
Neal
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The boundary of an $n$-manifold is an $n-1$-manifold

The following problem is from the book "Introduction to topological manifolds". Suppose $M$ is an $n$-dimensional manifold with boundary. Show that the boundary of $M$ is an $(n-1)$-dimensional manifold (without boundary) when endowed with the…
Saal Hardali
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Stokes' Theorem general case

With the following lemma : Lemma : Let $f_{+},f_{-} : U \longmapsto \mathbb{R}$ be $C^{1}$ maps with $f_{-} \leq 0 \leq f_{+}$ with $U \subset \mathbb{R}^{n}$ open and bounded with $C^{1}$ boundary. Let $\Omega : \lbrace (x,t) : x \in U \wedge…
jacopoburelli
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Is the square pyramid a manifold with corners?

An n-manifold with corners is topologically an n-manifold with boundary, but with a smooth structure that makes it locally diffeomorphic to $[0,\infty)^n$ instead of $[0,\infty) \times \mathbb{R}^{n-1}$. See also: J. Lee, Introduction to Smooth…
Friedrich
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Can you hear the pins fall from bowling game scores?

Let $\mathbb T=\{1,\dotsc,10\}$ represent the ten pins in a standard game of bowling. Given two sets of pins $T\subseteq S\subseteq \mathbb T$, let's write $p_{S\to T}$ to represent the conditional probability that given the current pins up are $S$,…
A. Rex
  • 1,710
16
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1 answer

Showing that $\bar{\mathbb{B}}^n$ is a manifold with boundary (Lee ITM Probelm 3-4)

"Show that every closed ball in $\mathbb{R}^n$ is an $n$-dimensional manifold with boundary, as is the complement of every open ball. Assuming the theorem on the invariance of the boundary, show that the manifold boundary of each is equal to its…
Duncan Ramage
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15
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2 answers

Constant Rank Theorem for Manifolds with Boundary

I'm trying to answer problem 4-3 from Lee's Introduction to Smooth Manifolds, 2nd edition. The problem says: Formulate and prove a version of the rank theorem for a map of constant rank whose domain is a smooth manifold with boundary. Lee himself…
Hempelicious
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(Whitney) Extension Lemma for smooth maps

I am currently reading Lee's book "Introduction to Smooth Manifolds (2nd edition)". Corollary 6.27 in that book states that a smooth map $f\colon A \rightarrow M$ where $M$ is a smooth manifold without(!) boundary and $A \subset N$ is closed (where…
PhoemueX
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1 answer

Euler characteristic of a manifold is odd

This was a past exam question: Let $M$ be a compact connected orientable topological $n$-manifold with boundary $\partial M$ so that $H_*(\partial M;\mathbb{Q}) \cong H_*(S^{n-1};\mathbb{Q})$. If $n \equiv 2$ mod $4$, show that the Euler…
xy15
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The definition of smooth maps given in Introduction to Smooth manifolds by John M. Lee

I'm currently reading Introduction to Smooth Manifolds by John M. lee. I'm trying to understand his definition of smooth maps $F:A\subseteq M\to N$ given in page 45. Let's start from scratch to have some context. Definition 1. If $A\subseteq…
Zero
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Codimension $1$ Embedding into $\mathbb{R}^{n+1}$

I am trying to determine which homotopy types can be realized by $n$-manifolds that have codimension one embeddings into $\mathbb{R}^{n+1}$. Suppose I have $X^n \subset \mathbb{R}^{n+1}$ and an embedded framed disk $D^k\subset \mathbb{R}^{n+1}, k <…
user39598
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10
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Tangent space to a surface at boundary points

Let $M$ be a $2$-dimensional compact oriented surface in $\mathbb R^3$ with boundary $\partial M$. For any $p\in M \setminus \partial M$ tangent vectors are defined as speed vectors of smooth curves $\Gamma \colon (-1,1) \to M$, $\Gamma(0) = p$ at…
Appliqué
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10
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2 answers

Non-surjective map implies zero degree for manifolds with boundary

Let $M$ and $N$ be compact, connected, orientable, topological $n$-manifolds with non-empty boundary, and let $$f : (M, \partial M) \to (N , \partial N) $$ be a continuous map of pairs. Then, we can define the degree of $f$ as the integer $d \in…
S.T.
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2 answers

Orientable manifold $M$ ,then $\partial M$ is orientable

Let $M$ a topological manifold of dimension $n$ with boundary $\partial M$. We define $M$ to be orientable if $M- \partial M$ is orientable. Here when I say orientable, I mean there is a locally coherent choice $\mu_x$ of generators of $H_n(M,M-x)…
Tommaso Scognamiglio
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9
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1 answer

Find sequential orthographic projections, linking three different manifolds of dimension $n=1,2,3$

Below is an image of the family of 2D curves for reference. The image is arbitrary. Still trying to formulate a concise question. I know that the geodesics for flat Euclidean Space are straight lines. But I want to take these curves and try to work…
J. Zimmerman
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