# 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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### Topological boundary as a submanifold

Let $U$ be an open subset of a smooth $n$-manifold. Consider $\partial U$ the topological boundary of $U$. Is the following true ? : If $\partial U$ is a smooth $n-1$ submanifold without boundary, ...
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### Is $\overline{u^{-1}((-\infty,a))}$ a $C^k$-manifold with boundary for a $C^k$ function $u$?

Let $u\in C^k(\mathbf{R}^d)$, $M:=\overline{u^{-1}((-\infty,a))}=\overline{\{x\in\mathbf{R}^d \mid u(x)<a\}}$, where $a$ is a constant such that $M\neq\emptyset$. Is $M$ a $C^k$-manifold with ...
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### Induced orientation on manifold

Suppose $M$ is a naturally oriented $n$-manifold embedded in $\mathbb{R}^n$ with non-empty boundary. Give $\partial M$ the induced orientation. The induced orientation defines a unit normal vector ...
Let $M^{k}$ a submanifold, $h:U\to M^{k}$ a chart, and $\gamma:[a,b]\to h(U)\subset M^{k}$ a curve in $M^{k}$. Represent the curve in coordinates $(h,U)$ as $h^{-1}\circ\gamma(t)=(y^{1}(t),...,y^{k}(t)... 0answers 66 views ### Show that a given set is a manifold with boundary given$A:= \{ (x_1,x_2,x_3) \in \mathbb{R} : x_1^2 + x_2^2 + x_3^2 = 18, x_3 \leq 2\}$show that$A$is a manifold with boundary and calculate$ \delta A$where$\delta A$is the boundary of A. I ... 0answers 82 views ### A$k+1$Manifold whose boundary is the solution set to the equation$f(\vec{x})=\vec{0}$Let$f: \mathbb{R}^{n+k} \to \mathbb{R}^n$be of class$C^r$. Let$f_1, ..., f_n$be components of$f$. Define $$M=\{\vec{x} | f(\vec{x})=\vec{0}\},$$ $$N=\{\vec{x} | f_1(\vec{x})=0, ..., f_{n-1}(\... 0answers 192 views ### 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 ... 2answers 108 views ### Are \mathbb{C}^2 and \mathbb{C}^2/(x,y)\sim(y,x) homeomorphic? Let A be the set of monic quadratics over \mathbb C and let B be the set of unordered pairs over \mathbb C where possibly the two elements of the pair may be the same. Then the map which takes ... 0answers 46 views ### Show that \partial(M\times N)=M\times\partial(N) Let M a k-dimensional manifold without boundary of \mathbb{R}^{n} and N a l-dimensional manifold of \mathbb{R}^{m} with or withour boundary. Show that \partial(M\times N)=M\times\partial(N) ... 2answers 119 views ### Show that M\#\mathbb S^n\approx M. I recall that M_1\#M_2 is the connexe sum of two manifolds and it's defined as following: Let B_1\subset M_1\backslash \partial M_1 and B_2\subset M_2\backslash \partial M_2 where M_i have ... 1answer 195 views ### Open neighborhood of a manifold boundary point Manifold with boundary: An n-dimensional manifold with boundary is a second countable Hausdorff space in which every point has a neighborhood homeomorphic either to an open subset of \mathbb{R^n}... 1answer 41 views ### In finding boundary of the product of two half-lines, shall homeomorphism be global? Lets \mathbb{R_{+}^{n}} = \mathbb{R^{n-1}} \times [0;+\infty[ Basically in my course I have this statement within the definition of a manifold with boundary: \forall x \in M, \exists U_x an ... 1answer 335 views ### The corner of the squre A square is a topological manifold with boundary but not a smooth manifold with boundary because of its corners. But I am confused about it. I think since for a specific corner p, there is only one ... 1answer 133 views ### Topological Manifold is Manifold with Empty Boundary I want to show that every n topological manifold M is an n Manifold with boundary where \partial M=\emptyset. i.e. every chart (U,\phi) maps to an open set V\subseteq\mathbb{H}^{n\circ} (... 1answer 25 views ### The nature of components in a certain manifold Let N be a smooth, connected manifold and f:N \to \mathbb R a smooth, proper and surjective map, transverse to some k \in \mathbb N. This means that M:=f^{-1}(k) \subset N is a finite ... 1answer 42 views ### Tangent vectors in T_p\partial M I know that if M is a smooth n-dimensional manifold with boundary, then \partial M is a smooth (n-1)-dimensional manifold. So for p\in\partial M, we have T_p\partial M\cong\mathbb{R}^{n-1}\... 1answer 53 views ### Computing parametrizations for a differentiable 2-manifold with boundary Consider the following subset of \mathbb{R}^{3} $$C=\{(x,y,z)\in\mathbb{R}^{3}\:|\:0\leq x\leq 1,\:0\leq y\leq 1,\:z=x^{2}+y^{2} \}.$$ Intuitively, this looks like a ... 2answers 1k views ### Topological Boundary vs Manifold Boundary Let A be the open unit disc in \mathbb{R}^2 and B be the closed unit disc in \mathbb{R}^2. The toplogical boundary of A and B is S^1. This I understand. The manifold boundary of A ... 0answers 32 views ### About the number of minimum parametrizations of a 1-smooth manifold compact w/ boundary in \mathbb{R}^{3} Let C be a 1-dimensional compact differentiable manifold with boundary in \mathbb{R}^{3}. In easy examples, it looks like we can always parametrize such a manifold with only two charts: usually ... 1answer 291 views ### Universal cover of boundary Let M be a compact manifold-with-boundary and B a component of \partial M. Let \tilde{M} be the univeral cover of M with infinite-sheeted covering map p:\tilde{M} \to M. I wonder about the ... 0answers 23 views ### property of sum of coefs of a chain Suppose c is a k+1 chain in U(open set in space R^n), then boundary of c (a k chain) can be expressed as a linear combination of k-cubes, using boundary operator:$$∂c=∑_ia_ic_i$$, where a_i are the ... 1answer 828 views ### Euler Characteristic of a boundary of a Manifold I need some guidance in understanding a specific passage of the following result taken from [tom Dieck Algebraic Topology, page 456] Proposition 18.6.2. Let B be a compact (n+1)-manifold with ... 1answer 31 views ### How to qualify a N dimensional manifold as Compact under following condions? Suppose a manifold of N dimensions is closed and bounded in a dimension but it remained unbounded in all other dimensions, so how to categorize the manifold. For example, in simpler form how to ... 3answers 79 views ### Is there any shorter term for manifolds with boundary? The “with boundary” does get a bit unwieldy when you have to write it more than a couple of times. I can't seem to find any alternative term on Wikipedia or elsewhere, but surely someone ... 1answer 61 views ### Reference on manifolds with boundary I am here because I want to know if someone knows of some good e fast books or references about manifolds with boundary. Help me please. 0answers 75 views ### Charts in an oriented manifold with boundary Let M be an oriented manifold (with boundary) with dim (M)\ge 2. Show that there exists an atlas \{(U_{\alpha},\phi_\alpha)\}_{\alpha\in I} for the chosen orientation such that \forall\... 1answer 282 views ### Tangent space of manifold has two unit vectors orthogonal to tangent space of its boundary I'm reading spivak calculus on manifolds and got stuck. Let M be a k-dimensional manifold with boundary in \mathbb{R^{n}}, and M_{x} is the tangent space of M at x with dimension k, then \partial ... 0answers 305 views ### Integration by parts on manifold with a boundary Suppose C is a 3-form, and G is a 4-form defined by G = dC. Also, M_{11} is an 11-dimensional manifold (without a boundary), W_{6} is a 6-dimensional submanifold of M_{11} and D_{\epsilon}... 0answers 88 views ### Reference for theorems in Hirsch In Hirsch's differential topology, we find the following theorems on page 31: 4.1 Theorem: Let M be a C^r \partial-manifold and N a C^r manifold, r \geq 1. Let f : M \to N be a C^r ... 1answer 449 views ### Basic examples topological manifolds with boundary I've just started to study differential geometry and I've some problems with the first definitions. We have defined a topological manifold with boundary of dimension n as a topological space M such ... 1answer 155 views ### When do we call the boundary \partial\Omega of a bounded domain \Omega\subseteq\mathbb{R}^n smooth? When do we call the boundary \partial\Omega of a bounded domain \Omega\subseteq\mathbb{R}^n smooth? I can't find a formal definition. I know, that we say, that \partial\Omega has a C^k-... 0answers 158 views ### Closure of a Manifold is a Manifold with Corners? Is there a general theorem that shows that if you have a manifold S then its closure \overline{S} is a manifold with corners? I am dealing with a specific set S (I would rather not say which ... 0answers 373 views ### Stokes' theorem: Induced orientation on the boundary of a manifold The Question Let K = \{(x,y,z) \in \mathbb{R}^3 : x^2 + y^2 + z^2 \geq 1\}, where K is oriented via the canonical volume form on \mathbb{R}^3: dx \wedge dy \wedge dz. Let \mathbb{S}^2 be ... 0answers 92 views ### Cartan geometry on manifolds with boundary I was reading Sharpe's text on Cartan geometry, and I started to wonder: Does the theory change in any significant way if the base manifold for the Cartan principal bundle is allowed to be a smooth ... 1answer 448 views ### Boundary of the boundary of a manifold with corners A point of a manifold with corners is a boundary point by definition if one of its coordinates is 0 by some (hence in all) chart with corners (see here). In the same page one can read: The ... 1answer 313 views ### Boundary of a topological manifold invariant? Let M=(X,\tau) be a topological manifold with boundary. One can proof that the interior Int(M) and boundary \partial M of the manifold are distinct sets. I was wondering if someone knows a ... 1answer 51 views ### Smooth mapping between manifold such that \text{Im}(f) \subset \partial N Let f:M \to N be smooth such that \text{Im}(f) \subset \partial N. Prove that f as mapping f:M \to \partial N is smooth. I've tried to write down f:M \to \partial N as composition of two ... 1answer 1k views ### What does it mean for a manifold to be oriented? I'm currently working through Spivak's Calculus on Manifolds. I've got to Stokes' Theorem, which is stated thus (the bold is my emphasis): Stokes' Theorem If M is a compact oriented k-... 0answers 122 views ### maximum principle on compact manifolds with boundary Let us consider the equation Lu + f(u) = 0 on a compact manifold \overline{M} = M \cup \partial M with boundary, with Dirichlet boundary conditions. L is a linear elliptic operator, and f ... 1answer 156 views ### Differential-form version of Cauchy-Schwarz on manifold boundary Consider a manifold M \subset \mathbb{R}^2 with boundary \partial M. Then, consider a zero-form field, \phi^{(0)}, defined on the whole of M (including the boundary). Then, the following holds:... 1answer 100 views ### G is an (n-1)-manifold without boundary and is the topological boundary to an open K\subset \mathbb{R}^n. Prove G \cup K is an n-manifold. All manifolds are smooth. Let M = G \cup K. The interior of M is an open set in \mathbb{R}^n and can be given a global coordinate by the identity map. The points in M not on the interior of ... 2answers 333 views ### Invertibility theorem on the boundary for a function between two closed 2D manifolds Assume a function f:\mathbb{R}^2\to\mathbb{R}^2 on a simply connected, closed domain D\subset\mathbb{R}^2 including its boundary \partial D. I am interested in the local invertibility of f ... 1answer 158 views ### Boundary points of a manifold I'm reading about Riemannian Geometry and my question is regarding Manifolds with Boundary. I want to show a point of a manifold with boundary is either an interior point or a boundary point, so no ... 1answer 757 views ### Is the complex projective plane a compact manifold with or without boundary (closed manifold)? my question is the one in the title. (My motivation is to understand in which way Freedman's classification of compact simply-connected 4-manifolds implies the Poincare conjecture for 4-manifolds, as ... 2answers 739 views ### Gauss-Green Theorem from generalized Stoke's Theorem. I am trying to deduce the next identity (Green-Gauss theorem)$$\int_\Omega \dfrac{\partial u}{\partial x_i} dx = \int_{\partial \Omega} uv_i dS$$from the generalized Stoke's theorem for manifolds. ... 1answer 193 views ### rudin's principles of mathematical analysis 10.31 I'm working on rudin's principles of mathematical analysis(3rd edition). There is problem too complex for me to solve.Please help me. The problem is on p270-271 of text. "Let T be 1-1 mapping of$Q^n$... 1answer 146 views ### The space of collars of a manifold is contractible Theorem: Let$M$be a smooth manifold with boundary$\partial M$. Let$e_0,e_1 : \partial M\times [0,1]\rightarrow M$be collars of$M$, i.e.$e_i$are embeddings such that$e_i(x,0)=x$for each$x\in ...
Let $(M,g)$ be a $d$-dimensional smooth, riemannian manifold without boundary, which is geodesically complete, connected, has positive injectivity radius and a bounded geometry (i.e. all derivatives ...