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

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20 views

What is the connection between $\sqrt g$ and $|\det \psi'|$?

My text defined integration on a manifold as follows Let $M\subset \mathbb R^n$ be an $m$-dimensional manifold, $\varphi:U\to V$ a local map $(U\subset\mathbb R^m, V\subset M)$ and $f:M\to\mathbb ...
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45 views

Sequence of critical values has no cluster point (Milnor, Morse Theory)

The following Claim is used in the proof of Theorem 3.5 in John Milnor's "Morse Theory": Claim: Let $f: M \rightarrow \mathbb{R}$ be a differentiable function on a manifold $M$ with no degenerate ...
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42 views

Find the maximal integral curve $c(t)$ starting at the point $(a,b) \in \mathbb{R}^2$ of the given vector field.

Yet another integral curve problem. The vector field this time is $X_{(x,y)} = \dfrac{\partial}{\partial x} + x \dfrac{\partial}{\partial y}$. So, using what I learned from my last post, I should ...
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1answer
51 views

Find the integral curves of the given vector field.

The vector field is as follows: $X_{(x,y)} = x \dfrac{\partial}{\partial x} - y \dfrac{\partial}{\partial y} = \begin{bmatrix} x \\y \end{bmatrix}$. I know that to find integral curves, you need to ...
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1answer
30 views

General Linear Group is(not) compactly generated

We know that any connected Lie Group is compactly generated. I have a feeling that $\mathbb{GL}_n\mathbb{R}$ is not compactly generated. Is it true? If it is how can I prove this? If not, what ...
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1answer
53 views

Does the connected sum depend on direction of gluing?

The connected sum of two surfaces (2-manifolds) is defined by removing a disk from each and gluing the cut edges: (Image adapted from Wikipedia) Does the resultant surface (up to homeomorphism) ...
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1answer
171 views

Definition of Lie Groups

In the definition of Lie Group, we require that $$(x,y)\rightarrow x*y \text{ and } x\rightarrow x^{-1}$$ both be smooth. Are there any examples of groups that satisfy only one of these and not the ...
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32 views

Real analytic manifold

I found in some lecture notes such a definition of real analytic manifold: Let $X$ be a complex manifold (ringed space, locally isomorpic to...), $i: X \rightarrow X$ - a conjugation ...
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1answer
45 views

Reference for introductory Lie Groups

I am currently learning about Lie groups,So kindly suggest a reference for Lie groups, which contains lecture on Manifolds as well.
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2answers
130 views

Closed, orientable surface whose genus is very hard to find intuitively

I'm introducing the Classification Theorem on closed and orientable surfaces in a talk on (intuitive) topology, and to motivate it I'd like an example of an embedding of a surface in $\mathbb{R}^3$ ...
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1answer
80 views

Calculus on manifolds

I am studying manifold theory ( smooth manifold). As I study the subject, I feel more and more confused. I am not yet at chapter dealing with deferential forms and integration. I can understand that ...
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50 views

$X \in T_pM$, there is a smooth vector field $\tilde X$ on $M$ such that $\tilde X_p=X$

I am trying to prove the lemma from the Lee, Introduction to Smooth Manifolds book: Lemma 3.16. Let $M$ be a smooth manifold. If $p \in M$ and $X \in T_pM$, there is a smooth vector field $\tilde X$ ...
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1answer
43 views

Identifying tangent space of manifold with set

Identify $ \mathbb{R}^4$ with the space of $2×2$ matrices $M(2×2,\mathbb{R})$. The set $M$ of matrices with determinant $3$ is a smooth manifold of dimension $3$. ...
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50 views

Constructing Riemann surfaces

At the risk of asking a question that has been already answered, I have been trying to figure out how to construct the Riemann surface of slightly more complicated examples, but after reading examples ...
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1answer
45 views

Isometry of spheres/hypersurfaces and more generally Riemannian manifolds.

Let $M$ and $N$ be two spheres (of different radius) in $\mathbb{R}^n$ of dimension $n-1$. Suppose there is a Riemannian isometry between them (so a diffeomorphism and isometry). Then distances must ...
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0answers
41 views

Proof of Euler Characteristic for Sphere

Theorem 1. All cell decompositions of a sphere $S$ have Euler characteristic 2. This is well-known, but I had this idea for an intuitive proof: for any cell decomposition $\Gamma$ with $V$ ...
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1answer
33 views

Second fundamental form without orientability?

Let $F$ be a $C^2$-hypersurface (or $n$-manifold) embedded in $\mathbb{R}^{n+1}$. Suppose $F$ is not orientable. Since I cannot choose a continuous global normal field, what consequences does this ...
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1answer
35 views

Canonical isomorphism between complexified tangent space of submanifold fixed by antiholomorphic involution and tangent space of complex manifold

I haven't really studied complex manifolds and I am at a bit of a loss in regards how to approach this problem: Let $M$ be an $n$-dimensional complex manifold, and let $\phi:M\rightarrow M$ be an ...
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1answer
60 views

Definition of smooth maps between manifolds

Here is a page from Guillemin-Pollack's differential topology: My question is: At the bottom he defines $df=d\psi\circ dh\circ d\phi^{-1}$. Why doesn't he just define $df=dh$, like here: ...
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36 views

Upper-half space of a manifold with boundary

Suppose that $X$ is a manifold with boundary and suppose that $x$ is a boundary point. Define the upper half space $H_x(X)$ in $T_x(X)$ to be the image of $\mathbb{H^k}$ under $d \phi_0:\mathbb{R^k} ...
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2answers
131 views

Expression for Laplace-Beltrami on sphere?

Is there a good expression for the Laplace-Beltrami on a function $u$ on a sphere or a circle of radius $R>0$ in terms of the Laplacian on ambient space? There is a formula on Wikipedia for the ...
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33 views

Example of 1-dimensional hypersurface in $\mathbb{R}^2$ which is compact?

Is there an explicit example of a $1$-dimensional $C^k$ hypersurface in $\mathbb{R}^2$ which has no boundary and is compact? I know of a circle, but want something like an interval.
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22 views

Does this map define a Smooth Immersion & Embedding

I need help showing the following is a smooth embedding. $B:(R/Z)->S^2$ defined by $B([t])=(cos(t),sin(t),0)$ I know that this map is not a submersion because it never vanished and the ...
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1answer
45 views

Smooth Embedding

I need help showing that the following smooth map is not a smooth embedding: $f:\mathbb{S}^1 \to \mathbb{R}$ defined by $f(z)= \operatorname{Re}(z)$. I know that this map is not a submersion because ...
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77 views

If there exists a diffeomorphism between two surfaces, what is the relation between Laplace-Beltrami operators on the surfaces?

Let $S(0)$ and $S(t)$ be a hypersurface in $\mathbb{R}^n$. Suppose there is a diffeomorphism $F^0_t:S(0) \to S(t)$. Suppose we have the Laplace-Beltrami operator $\Delta_{S(\cdot)}$. Let $u:S(t) \to ...
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1answer
14 views

If $S$ is a $C^k$ hypersurface, is $S\times (0,\infty)$ a $C^k$ hypersurface too?

Let $S$ be an $n$ dimensional $C^k$ hypersurface in $\mathbb{R}^{n+1}.$ Is $S \times (0,\infty)$ also a $C^k$ hypersurface (in $\mathbb{R}^{n+2}$)? I don't know what the chart map should be...
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0answers
62 views

Prove that a map is a homeomorphism and the inverse is bounded

I'm trying to unravel an obscure passage in a textbook, which states that if $\phi :\mathbb{R}^m\to\mathbb{R}^m$ is continuous, bounded and Lipschitz with constant $\varepsilon$ (which is still free ...
3
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1answer
46 views

submanifold of Euclidean space is oriented if and only if normal bundle is an oriented vector bundle.

Let $f:M\longrightarrow \mathbb{R}^{n+k}$ be an immersion of $n$-dimensional manifold $M$ into $\mathbb{R}^n$. Let $\nu(M)$ be the normal bundle of $M$. Prove that $M$ is oriented if and only if ...
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0answers
22 views

A question on the completely positive maps and manifold structure

I was reading a paper in which the curvature and Euler characteristic of a completely positive map (in finite dimensions). Let \begin{equation} \Phi(X)=\sum_{j=1}^nV_jXV_j^* \end{equation} be a ...
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1answer
58 views

Passive and active coordinate transformation on a topological manifold.

Let us suppose we have $m$-dimensional smooth topological manifold $M$. Let $(U,\varphi)$ and $(V,\psi)$ be two charts on the manifold and $U \cap V \neq \emptyset$. For a point $p \in U \cap V$, we ...
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31 views

Antipodal map and parallel transport on $S^3$

I'm reading Thurston and Levy's "Three-dimensional geometry and topology" where they have an informal explanation of what life in $S^3$ would look like. For the following detail that I am concerned ...
2
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1answer
69 views

Tangent Space to a manifold

So, I have a manifold $M=\{\mathbf{x}:\mathbf{\Theta}\left(\mathbf{x}\right)=\mathbf{0}\}$. I can also write $M=\{\mathbf{x}:\mathbf{F}(\mathbf{x})=\mathbf{c}\}$. Both functions are differentiable. I ...
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39 views

finite-dimensional continuous vector bundle

Let $M$ a compact metric space and $\pi: F \rightarrow M$ a finite-dimensional continuos vector bundle over $M$, endowed with a continuous Riemannian metric. I was wondering if it will be true that: ...
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1answer
30 views

Endowing a metric on the torus from the euclidian metric of its covering space, the plane

In Thurston's and Levy's "Three dimensional Geometry and Topology, page 6, they define the induced metric on the torus from the euclidian metric of its covering space, the plane. Specifically, for ...
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1answer
38 views

Normal vector of $\Gamma \times \mathbb{R}^+$ where $\Gamma$ is compact hypersurface

Let $\Gamma$ be a smooth boundaryless hypersurface of dimension $n-1$ in $\mathbb{R}^n$. Define $Q=\Gamma \times \mathbb{R}^+$. What does a normal vector of $Q$ look like? Because I want to compute ...
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1answer
58 views

measure on non-oriented Riemannian manifold

Let $M$ be a non-oriented Riemannian manifold of dimension $m$. Nash embedding theorem implies that there exists an isometric embedding $\phi: M\longrightarrow \mathbb{R}^n$ for $n$ sufficiently ...
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1answer
33 views

Projection of a Manifold has measure zero

Let $M$ be a $k$-dimensional sub manifold of $\mathbb{R}^N$, and let $\pi_n:\mathbb{R}^N\to\mathbb{R}^n$ be the canonical projection, with $n>k$. Can we show that $\pi_n(M)\subset \mathbb{R}^n$ has ...
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1answer
43 views

Immersed submanifold

I need help finding an example that shows an immersed submanifold might have more than one topology and smooth structure with respect to which it is an immersed submanifold. This is problem 5-15 from ...
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1answer
71 views

Definition of fractional Laplacian on a compact manifold?

How does one define the fractional Laplacian operator $(-\Delta)^s$ on a compact Riemannian manifold? In $\mathbb{R}^n$, it is defined $$ (-\Delta)^s f(x) = c_{n,s} \int_{\mathbb{R}^n} \frac{f(x) - ...
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1answer
51 views

Tangent bundle of a manifold [duplicate]

can anyone help me with this problem: Show that for a manifold $M$, the tangent bundle $TM$ also has the structure of a manifold. If $M$ is an n-manifold, what is the dimension of $TM$? for the 1st ...
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1answer
68 views

Is every “nice” topological vector space a manifold?

Say $V$ is a topological vector space. What conditions do you need to add on $V$ to make it a (topological, maybe infinite-dimensional) manifold? For instance, can we view the Schwartz class ...
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1answer
19 views

Symmetry group of a non abelian group manifold

Reading an article ...$B$ is itself the manifold of some group $H$. It should be noted that, if $H$ is a non abelian group, the symmetry group $G$ of the group manifold is not $H$ but ...
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Circle action on the product of a Mobius band and a circle.

Consider the product of a Möbius band and a circle $Mo\times S^1$. Is there a circle action on $Mo\times S^1$ such that it is equivariantly homeomorphic to the twisted product $D^2 ...
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1answer
51 views

volume of some stochastic processes

for a continuous and differentiable curve $\vec{x}_t$ in $\mathcal{R}^n$ parameterized by a single variable $t$, there is a well defined way of computing the volume of this one-dimensional manifold ...
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44 views

Variational Problem on a manifold - bounding my functional from below

Let $(M, g_{ij})$ be a compact Riemannian manifold. Let $f \in C^{0,\alpha}(M)$ be a given function. I have the linear operator $L = \Delta h - 12u^2h$, where $u \in C^{2,\alpha}$. I need to show ...
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4answers
84 views

1-manifold is orientable

I am trying to classify all compact 1-manifolds. I believe I can do it once I can show every 1-manifold is orientable. I have tried to show prove this a bunch of ways, but I can't get anywhere. ...
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1answer
38 views

Are vector bundles isomorphic when their transition functions are homotopic?

I'm in the process of trying to understand the equivalence between two different approaches to vector bundles, namely defining them to be smooth surjections $E\to B$ that are locally isomorphic to ...
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0answers
53 views

Hatcher Theorem 3.26 - orientability

I am reading Hatcher, the beginning of the chapter on Poincare duality. I am trying to understand how theorem 3.26 is deduced from lemma 3.27 and I must admit I find Hatcher's proof very esoteric. ...
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34 views

why does the differential of this function on the manifold have this expression?

I am studying the proof of a theorem from Warner, where I am stuck in the following place : Let $\psi:M\longrightarrow N$ be an immersion and, let $m\in M$. Let $(W,\tau)$ be a coordinate system ...
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0answers
70 views

Convergence of a sequence pointwise a.e. on a manifold given that it converges on a reference manifold

Let $\Gamma_t$ be a compact hypersurface for each $t \in [0,T]$. Let $Q=\Gamma_0\times(0,T)$ and $$Q_T :=\bigcup_{t \in (0,T)} \Gamma_t \times \{t\}.$$ For each $t \in [0,T]$, suppose that ...