Tagged Questions

For questions on manifolds of dimension $n$, a topological space that near each point resembles $n$-dimensional Euclidean space.

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Finding this transition function for homeomorphisms over a cylinder

Let $C = \left\{ \ (x,y,z) \in \mathbb{R}^{3} \ | \ x^{2} + y^{2} = 1,\ 0 \leq z \leq 1\ \right\}$ I've got the functions $f:(0,1) \times (0,1) \to C$ and $g:(0,1) \times (0,\tfrac{1}{3}) \to C$ ...
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It can happen that $M_1 \cap M_2 =\emptyset$ but $M_1 \cup M_2$ is not a $k$-dimensional manifold. Give a counter example.

Let $M_1,M_2 \in \mathbb{R}^n$ be $k$-dimensional manifolds, $M=M_1 \cup M_2$ It can happen that $M_1 \cap M_2 =\emptyset$ but $M$ is not a $k$-dimensional manifold. Give a counter example. ...
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M is a k-manifold if and only if $\phi(M)$ is a k-manifold

Let $\phi: \mathbb{R}^n\rightarrow \mathbb{R}^n$ be a diffeomorphism and $M\subset \mathbb{R}^n$ M is a k-manifold if and only if $\phi(M)$ is a k-manifold. Prove it. So what I did was try to ...
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What do we need to guarantee that $[X, Y]_p$ is linearly independent with $X_p$ and $Y_p$?

I am trying to figure out the conditions such that $[X, Y]_p$ is linearly independent with $X_p$ and $Y_p$ for some vector fields $X, Y$ and some $p$ in a three-dimensional manifold. I have that ...
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Extending covering projection of the boundary

Let $M$ and $E$ be (topological) manifolds with boundaries $\partial M$ and $\partial E$ respectively and assume we have a finite-sheeted covering $\rho: \partial E\to \partial M$. Is it possible to ...
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Jacobian of a diffeomorphism

Let $U,V\subseteq \mathbb{R}^{n}$ be open. Let $\alpha:U \to V$ be a smooth homeomorphism. Furthermore, assume that $\mathcal{J}_{\alpha}(\mathbf{x})$ (the Jacobian matrix) has rank $n$ for all ...
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Orient Manifold

$\mathbf{Problem \,2.}$ Consider the $2$-manifold in $\Bbb R^3$ given by $$x^2+y^2+z^2=1,\qquad z\ge 0.$$ Orient $M$ such that $\alpha$ in the Equation $(2)$ belongs to the orientation, and give ...
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Tangent Space Well Defined?

Question: Let $M$ be a $k$-manifold of class $C^r$ in $\mathbb R^n$. Let $p\in M$. Show that the tangent space to $M$ at $p$ is well-defined, independent of choice patch. Unsure if I'm ...
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Quotient space of a linear space space is also linear?

Suppose, $C$ is a linear manifold (i.e., manifold which is closed under addition and multiplication) and $\Gamma$ is a Lie group. Can we say in general $C/\Gamma$ is also a linear manifold? Can we ...
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Vector field on manifold

I've only seen a vector field $V$ on a manifold $M$ as a mapping $V:M\to TM$. Is it true that they can also be seen as a mapping $V:C^{\infty}\left(M\right)\to C^{\infty}\left(M\right)$? How would $V$ ...
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Proving that every $n$-submanifold of $\mathbb{R}^{n}$ has a natural orientation

Let's say that $\mathcal{M}$ is a smooth submanifold of dimension $n$, of $\mathbb{R}^{n}$. Using my definition, this means: for every point $p \in \mathcal{M}$ there exists a coordinate patch ...
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Smooth vs topological orientation

I am confused about the different notions for a closed smooth manifold to be oriented. In my mind there are several equivalent ones: 1) Coherent pointwise orientation of the tangent spaces. 2) ...
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If $\int_M \omega=0\Rightarrow \omega=d\varphi$, then $H^n_c(M)\simeq\mathbb{R}$? ($M$ is a connected orientable manifold)

I'm reading a book in wich the author uses this argumet the whole time. For example, he assumes that $\int_\mathbb{R}\omega=0$ then $\omega =df$ and then he concludes that that ...
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Do the level sets of a constant rank map give a foliation of the domain?

I'm working through Smooth Manifolds by Lee and came to problem 19-8, where you're asked to prove that the level sets of a submersion form a foliation of the domain. However when I try to solve this ...
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Describing the Unit Normal to a Cylinder in $\mathbb{R}^{3}$

This is problem 34.4 in Munkres' Analysis on Manifolds. Let $\mathcal{C} = \{ \ (x,y,z) \in \mathbb{R}^{3}\mid x^{2} + y^{2} = 1 \text{ and } 0 \leq z \leq 1 \ \}$. Orient $\mathcal{C}$ by declaring ...
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I have no idea what Differential Forms are… [closed]

So in my Calc 3 class we use Shifrin's "Multivariable Mathematics", and his discussion on Differential Forms and Integration on Manifolds is impossible for me to follow. Can someone recommend ...
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Formal proof that (x,|x|) is not a smooth submanifold of $\mathbb{R}^2$

I have perused the related questions on this site, and was unable to find a formal proof of the fact stated in the title. Essentially, I have two questions: Is it a fact that if $M$ is a ...
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Foliations vs Laminations

What's the big difference/similarity between foliations and laminations? What kind of theorems hold for both of them? Is there something that makes them essentially the same/different?
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Are there countably many closed manifolds in each dimension?

There is a single closed topological 1-manifold (up to, of course, homeomorphism): $S^1$. The classification of surfaces shows that there are countably many closed topological 2-manifolds. ...
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Is $T(M)$ necessarily equivalent to the normal bundle of $M$ in $\textbf{R}^{2n}$, if $M$ is totally real?

Consider real submanifolds, $M^n \subset \textbf{C}^n$. ($\textbf{C}^n \equiv (\textbf{R}^{2n}, J)$, where $J(x, y) = (-y, x)$). $M^n$ is totally real if $J(T_pM) \cap T_pM = 0$, for all $p \in M$. ...
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Embedding of $2$-torus minus one point into $\mathbb{R}^2$ as an open set?

Let $M^2$ be the $2$-torus minus one point. Is there an embedding of $M^2$ into $\mathbb{R}^2$ as an open set?
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Can $S^k \times S^n$ be embedded into $\textbf{R}^{k + n + 1}$? [duplicate]

As the question title suggests, can $S^k \times S^n$ be embedded into $\textbf{R}^{k + n + 1}$, i.e. in codimension $1$?
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Immersion of $M^n$ into $\mathbb{R}^n$, is $M^n$ orientable? Compact? [closed]

Say we have an immersion of $M^n$ into $\mathbb{R}^n$ (same dimension). I have two questions. Is $M^n$ orientable? Is $M^n$ compact? Thanks in advance!
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On the meaning of formal sums of $k$-cubes, i.e. $k$-chains (in integration on manifolds)

A singular $k$-cube in $A \subseteq \mathbb R^n$ is a continuous function $c : [0,1]^k \to A$. A singular $0$-cube in $A$ is then a function $f : \{0\}\to A$, what amounts to the same thing, a point ...
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Where to find about the category theoretic study of manifolds?

I'm looking for a resource about a category theoretic study of manifolds. What do you think is a good start? Hint: Not after very advanced resources. So no worries (indeed, preferred) if it's an ...
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What's the normal space of the manifold $z = x^2 + y^2$?

What's the normal space of the manifold $z = x^2 + y^2$? Let's say I have a continous function $g: M \rightarrow S^n$ that sends every point on the manifold to the unit normal vector. I want to know ...
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Show the parametrized torus is a 2-dimensional smooth submanifold of$\mathbb{R}^3$ [duplicate]

How can I show that the parametrized torus $T=\{(x,y,z)\in \mathbb{R}^3 : (\sqrt{x^2 +y^2}-a)^2 +z^2 =b^2 \}$ is a 2-dimensional smooth submanifold of $\mathbb{R}^3$ ? I was thinking of using the ...
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Why is $f(U) \cap V$ the zero set of $y^{n+1}, \dots, y^m$? [duplicate]

I'm studying Tu's proof (p. 123) of theorem 11.13, but I just have a question about one detail. He has that $f\colon N\to M$ is an embedding of a manifold of dimension $n$ in a manifold of dimension ...
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Let $f : A\subset \mathbb{R}^{n+1} \to \mathbb{R}$, what does mean that $f$ is a submersion?

I am trying to answer the following question: Let $M_a := \{ (x^1,\ldots,x^n,x^{n+1}) \in \mathbb{R}^{n+1} : (x^1)^2 + \cdots +(x^n)^2 - (x^{n+1})^2 = a\}$. For which values of $a$, $M_a$ is a ...
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Computing Sectional Curvature on Hyperbolic Plane

For a pair of $(X,Y)$ of linearly independent vectors in $T_pM$, $p\in M$, the sectional curvature is defined as $$K_p(X,Y)=\frac{<R(X,Y)Y,X>}{|X|^2 |Y|^2 - <X,Y>^2}$$ The problem I'm ...
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Equivalent definition of properly discontinuous action

In the book An Introduction to Differentiable Manifolds and Riemannian Geometry by Boothby in Chapter $3$ the author gives the following definition: Definition($8.1$) A discrete group $\Gamma$ ...
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Uniqueness of dimension of regular submanifold

Suppose $N$ is a manifold of dimension $n$. Now a regular submanifold $S$ of $N$ of dimension $k$ is defined as, if for every point $p$ of $S$ there is a coordinate chart $(U,u_*)$ from a maximal ...
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Proof for showing that a set of space curves form a manifold

I basically have a smooth space curve $\alpha$,with curvature $\kappa$ and $\tau$, both non-zero, and I generate a family of curves $M_{\alpha} = \{\dfrac{\alpha}{\mu} : \mu \in (0, \infty) \}$ . The ...
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Trajectories of vector fields on compact manifolds

Suppose that $X$ is a smooth vector field on a smooth manifold $M$. The trajectories of $X$ are curves $p(t)$ in $M$ which satisfy $d{p(t)}/{dt} = X(p(t))$. It's well known that $p(t)$ exists ...
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Proving that Levi-Civita connection is preserved by isometries

I am trying to prove that given two Riemannian submanifolds $S,S'$ with Levi-Civita connections $\nabla , \nabla'$ and an isometry $f$, then $$Df(\nabla_XY)=\nabla'_{X'}Y'$$ where, ...
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How to interpret the cotangent bundle of a complex manifold?

Let $X$ be a complex manifold. I am not sure what people mean when they talk about the cotangent bundle $T^*X$ of $X$. I have two interpretations: At each point $x\in X$, $T_x^*X$ is the complex ...
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Do all continuous real-valued functions determine the topology?

Let $X$ be a topology space. If I know all the continuous functions from X to $\mathbb R$, will the topology on $X$ be determined? I know the $\mathbb R$ here is, somewhat, artificial. So if this is ...
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Why this two surfaces have one end?

I want to prove that the infinite-holed torus and the infinite-jail cell window have one end but the doubly infinite-holed torus doesn't, my definition of one end is the following: A locally ...
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How to interpret $a_i^k g_{kj}g^{jl}$? (Tensor index notation)

I have the equation $$a_i^k g_{kj}=-b_{ij},$$ where $a_i^k$ are given by: $a_i^k \cdot\vec r_k=\vec n_i$. Here, $g_{ij}$ and $b_{ij}$ are the matrices that determine the first and second fundamental ...
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The boundary $\partial S$ of the square in $\mathbb{R}^2$ has no topology and smooth structure which makes it an immersed submanifold.

This is a problem in Smooth Manifolds by Lee, but it seems like there is an obvious decomposition of the $\partial S$ into a manifold with 4 components. The top and bottom edges including the corners ...
Two atlases on a manifold $M$ are equivalent if and only if they determine the same set of smooth functions $f:M\rightarrow\mathbb{R}$
Suppose $\{\phi_\alpha\}_{\alpha\in\mathcal{A}}$ and $\{\phi_\beta\}_{\beta\in\mathcal{B}}$ are two smooth atlases on a topological manifold $M$. My definition of two such atlases being equivalent is ...
Here is my view of orientability on a vector space $V$ of dimension $m>0$: let $I(V)$ be the set of linear isomorphisms from $V$ to $\mathbb{R}^m$. Given $\rho,\sigma\in{I(V)}$, we get a linear ...