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

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Isomorphism of the Clifford bundle of a Riemannain manifold

Let $(M,g)$ be an oriented Riemannian manifold and $Cl(M):=\bigcup_{x\in M}Cl(T_xM,g_x)$ be the clifford bundle of $(M,g)$. (Here $Cl(T_xM,g_x)$ denotes the clifford algebra of the vector space ...
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1answer
55 views

For $(n-1)$-form $\omega$ on $M^{n}$ compact, orientable without boundary, then $d\omega$ vanish for some point

Let $M^{n}$ manifold compact, orientable without boundary and $\omega$ $(n-1)$-form then there is $p\in M$ such that $d\omega(p)=0$. This is for my homework of integration on manifolds & Stokes ...
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1answer
46 views

Eigenvalues of a Plane Curve Laplace-Beltrami Operator

Given a closed plane curve $C$, which is a one-dimentional manifold, what are the eigenvalues of Laplace-Beltrami operator defined on this curve? I know that the LB eigenvalue problem for unit ...
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1answer
26 views

Show that the tangent space of a manifold is a certain set.

Let $A\subset \mathbb{R}^n$ an open set, and $g:A\to \mathbb{R}$ continously differentiable such that $g'(x)\not=0 $ for $x\in A$. If $M = g^{-1}(\{0\})\not=\emptyset$, then I want to show that the ...
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76 views

Higher homology groups of infinite cyclic cover

Prove that all homology groups of the infinite cyclic cover of a knot complement are trivial except $H_1$. I've posted an answer below using Mayer-Vietoris. If you know of other arguments, please ...
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2answers
74 views

Why the set $g^{-1}(\{0\}) $ is not a differentiable manifold?

Let $g:\mathbb{R}^2 \to \mathbb{R}$ given by $g(x,y) = x^2 - y^2$. Then I am triying to figure out why this function is not a differentiable manifold , I was trying to give an explicit coordinate ...
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4answers
63 views

Are there compact manifolds without boundary?

Based on this question I'd like to know: Are there compact (sub)manifolds without boundary in $\mathbb{R}^n$ ? Because, as that question shows, the topology of the manifolds has to be the trace ...
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19 views

Pipe-fitting conditions in 3D

Let's we have smooth (continuous and infinitely differential) curve $f(x(t), y(t), z(t)) = 0$ in 3D. Now I want to build a pipe of diameter $D$ around it. Questions: What are the set of conditions ...
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75 views

Tautological line bundle over $\mathbb{RP}^n$ isomorphic to normal bundle? Also “splitting” of transition functions

Hallo fellow mathematicians. I try to understand why the normal bundle of $\mathbb{PR}^n$ is isomorphic (in the category of vector bundles) to the tautological line Bundle. More aptly, why ...
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What does it mean for a function f to be C^k “close” to a function g?

My impression is that $f$ is $C^k$ "close" to $g$ if $f$ is close to $g$, $f'$ is close to $g'$, ... $f^{(k)}$ is close to $g^{(k)}$. However, my professor is being a bit nebulous on what "close" ...
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36 views

Explaining problem in Gadea's “Analysis and Algebra on Differentiable Manifolds”

I have a lot of trouble trying to explain to myself what the author did in problem 1.102 (the answer is in the link): Let $TM$ be the tangent bundle over a differentiable manifold $M$. Let ...
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1answer
33 views

Codifferential and corresponding homology theory

This is the kind of a natural question which can come to mind after completing the standard course in differential geometry and homology theory: lety us start with a smooth manifold $M$. One can ...
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1answer
43 views

Nonorientable manifolds being a boundaries

I will say that a manifold (smooth, compact, without boundary) is itself boundary if there is some smooth compact manifold $W$ with boundary, such that $M=\partial W$. I'm interested in nontrivial ...
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1answer
43 views

Meaning of conditions on definition of topological manifold

This is my first time to study on the manifold. I've studied that the topological manifold is: Hausdorff, locally homeomorphic to Euclidean space, and second countable topological space. With ...
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1answer
43 views

Grassmanians and boudaries of manifolds

Let $M$ be a smooth, compact manifold without boundary. I will say that $M$ is a boundary when there is a smooth, compact manifold with boundary $W$ such that $\partial W=M$. After some lectures I ...
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1answer
37 views

G-P Exercise, immersion except at origin, what does its image look like?

(This is not a duplicate of another question on math.stackexchange, as that other question just basically asks for the answer to the question below, of which I have provided an answer to. My question ...
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Can the same surface have minimal genus in both a 3-manifold and a 4-manifold?

By a surface of minimal genus I mean in it's homology class: A surface $S_0$ embedded in a smooth manifold $M$ such that any other surface $S$ with $[S]=[S_0]\in H_2(M)$, we have $g(S)\geq g(S_0)$. ...
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1answer
41 views

Find a surface that has positive constant curvature that is not open subset of sphere

Can some one find a surface that has positive constant curvature that is not open subset of sphere. I know every connected and compact surface with positive constant curvature is sphere. I need ...
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1answer
27 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 ...
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1answer
94 views

Momentum a cotangent vector?

Imagine we have a particle described by $x \in M$, where $M$ is some manifold, then it is very intuitive I think that a velocity is an element of the tangent space at $x$, so $x' \in T_{x}M.$ Thus, by ...
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39 views

First exercise of Guillemin-Pollack. [closed]

If $k < l$ we can consider $\mathbb{R}^k$ to be the subset $\{(a_1, \dots, a_k, 0, \dots, 0)\}$ in $\mathbb{R}^l$. Show that smooth functions on $\mathbb{R}^k$, considered as a subset of ...
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1answer
270 views

How to show that the diagonal of $X\times X$ is diffeomorphic to $X$?

The diagonal $Q$ in $X\times X$ is the set of points of the form $(x,x)$. Show that $Q$ is diffeomorphic to $X$, so $Q$ is a manifold if $X$ is. Can anyone please help me to solve this question I ...
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20 views

Area between two curves as manifold with boundary

Let $U \subset \mathbb{R}^n$ be open set, $F,G:U \to \mathbb{R}$ smooth function such that $F(x)<G(x)$. We define: $$\Omega=\{(x,y) \in U \times \mathbb{R}:G(x) \leq y \leq F(x)\}$$ I would like ...
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21 views

global manifolds

Can you also explain why the global stable manifold is the union of the flow of the local stable manifolds for t < 0? Why do we not include all t?
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1answer
48 views

Let $M\subseteq \mathbb{R}^k$: Manifold topology vs. trace topology

I'm confused about the topology of submanifolds of $\mathbb{R}^n$: Let $M$ be such a $k$-manifold (say, the circle $S^1$, of dimension $1$, embedded in say $\mathbb{R}^7$); the topology of such a ...
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2answers
230 views

Topological manifolds (dimension)

I am taking an introductory course to topology and the professor defined a topological manifold of dimension $n$ if it is hausdorff and if for every point $x$ there exists an open set $U$ around $x$ ...
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21 views

About a map between two topological manifolds with different dimensions

Let $M_1$ be a $n$-dimensional topological manifold and let $M_2$ be a $m$-dimensional topological manifold, such that $m>n$. Moreover, let $U\subset M_1$ be an open set and let $f:U\rightarrow ...
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0answers
35 views

What can I say of an $m$-dimensional submanifold $S$ of an $m$-dimensional manifold $M$?

I consider a differentiable manifold $M$ of dimension $m$. Let be $S$ a submanifold of $M$ of the same dimension $m$. What can I say about $S$? I have tried to prove that $S$ is open but I get ...
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1answer
50 views

Quotiented manifold homeomorphic to a complex projective space?

I define an action on $\mathbb{C}-0 × \mathbb{C^2}-(0,0)$ by $(x,y,z) \mapsto ((1/a)x,ay,az)$ when $a$ is a non zero complex number, I get a manifold by quotienting. Taking element from this ...
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how to prove torus as a 2 dimensional manifold

Consider equation of torus in 3 dimension $(R-\sqrt{(x^2+y^2)})^2+z^2 = r^2$ where $R$ is larger radius and $r$ is smaller radius. how to prove that it is 2 dimensional manifold? I tried ...
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1answer
37 views

Prove that $X=\{(x,y,z,t) | x^2+6y^2+4z^2+t^2=1 \}$ is compact and connected

Let be $X=\{(x,y,z,t) | x^2+6y^2+4z^2+t^2=1 \}$. I have proved that $X$ is a submanifold of $\mathbb{R}^4$ of dimension $3$. I have to prove that $X$ is compact and connected. My idea, thinking of ...
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1answer
48 views

Integration with 2-forms

Wikipedia says: Let $$ \omega=f_{z}\, \mathrm dx \wedge \mathrm dy + f_{x}\, \mathrm dy \wedge \mathrm dz + f_{y}\, \mathrm dz \wedge \mathrm dx $$ be a 2-form on a surface with parametrization ...
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1answer
91 views

Is $S^1 \times S^1$ really a torus?

Consider a function $f(x)$ that is $2\pi$ periodic. Consider another function $g(y)$ that is also $2\pi$ periodic. If I wanted to compute the integral of either of these functions I would do so ...
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47 views

An example of 4D Hypersurface in 3D

Number of combinations of 4 dimensions choosing 3 at a time is 4. Someone please give a description of a most elementary 4 Dimensional Hyper surface which has its four 3D intersections with ...
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1answer
32 views

Prove that $\mathbb{R}^2 \times S^1 $ and $M=\left\lbrace (x,y,z,w) \in \mathbb{R}^4 \mid x^2+y^2- z^2-w^2 = 1 \right\rbrace$ are diffeomorphic

Let be $M= \left\lbrace (x,y,z,w) \in \mathbb{R}^4 \mid x^2+y^2- z^2-w^2 = 1 \right\rbrace$. I have proved that $M$ is a embedded submanifold of $\mathbb{R}^4 $ of dimension $3$. I have now to ...
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0answers
38 views

What is a real structure on a manifold?

I have been looking at manifolds (twistor spaces) that have a "real structure". I am not quite sure what this means. I've looked on Wikipedia and they have an article that explains real structures on ...
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66 views

Properties of the category of smooth vector bundles over a smooth manifold

I am wondering if there are any sources that discuss the properties of the category of vector bundles over a smooth manifold. It seems that most differential geometry texts I've looked at avoid ...
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1answer
107 views

How could a group be a manifold?

For example a Lie group is defined as a certain differentiable manifold, but what does this mean geometrically, and what is gained by viewing something abstract and algebraic as a manifold? First, I ...
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1answer
41 views

Retraction to the Boundary on Compact Manifold

I was given the following question on an exam today, "Suppose that $M$ is a compact $n$- dimensional oriented manifold with corners. A retraction to the boundary is a continuously differentiable map ...
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1answer
61 views

Is $C = \{(x,y) \mid x^3 + xy + y^3 = c \} \subset \mathbb{R}^2$ an embedded submanifold of $\mathbb{R}^2$?

The problem As a continuation of this question (where it was shown that $C$ was a closed $1$-dimensional submanifold for $c \neq 1/27$), I'm trying to find out whether or not $$C = \{(x,y) \mid x^3 + ...
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2answers
31 views

Properties of the ring of smooth function germs, question on proof.

Let us denote by $C_n$ the ring of $C^{\infty}$ smooth function germs $f : (\mathbb R^n, 0) \to \mathbb R$ or the ring of analytic functions germs $f : (\mathbb C^n, 0) \to \mathbb C$. Denote by ...
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1answer
70 views

How to show that $f : \mathbb{R}^n → \mathbb{R}^n$, $f(x) = \frac{h(\Vert x \Vert)}{\Vert x \Vert} x$, is a diffeomorphism onto the open unit ball?

Could anyone help me with the following problem? The problem Fix $\varepsilon \in (0, 1)$ and choose a smooth function $h$ on $[0,\infty)$ such that $h'(t) > 0$ for all $t ≥ 0$, $h(t) = t$ for ...
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1answer
79 views

Linear algebra revisited: What do we do when we set a coordinate system?

I was learning about covariant and contravariant vectors due to special relativity, and it occured to me that we don't live in $\mathbb{R}^4$. I'll explain myself better. Consider the space of ...
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1answer
35 views

Mean value theorem on Riemannian manifold?

Is there some generalisation of the classical mean value theorem for real-valued functions on an interval $$|f(x)-f(y)| \leq |\nabla f(c)||x-y|$$ for some $c$ between $(x,y)$ to the case where $f:M ...
2
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1answer
29 views

Adjoint representation and tangent vectors

Let $G$ be a Lie group, $\mathfrak{g}$ its Lie algebra, $\text{Ad}:G\rightarrow GL(\mathfrak{g})$ the adjoint representation of $G$. Then, for $X,Y\in \mathfrak{g}$, \begin{align*} ...
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1answer
33 views

Modelling the Möbius strip using implicit functions

While researching on Möbius strips I found its parametric representation on a lot of websites claiming it is easier. Can someone please explain what problems appear when modelling the Möbius strip ...
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1answer
51 views

Area form a differential form?

I just read that $\omega_x( \eta, \zeta) := \langle x, \eta \times \zeta \rangle $ is the area form on the sphere, where $x \in \mathbb{S}^2$ and $\eta,\zeta \in T_xM.$ All I see is that this is ...
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1answer
38 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 ...
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1answer
29 views

Quick question about covariant derivative

Let $f$ be a function and define $\nabla_X f = X(f)\,\,(1)$, where $\nabla$ is the connection on a manifold and as far as I understand the r.h.s is a function and $X$ is a vector field. I am just a ...
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1answer
48 views

Homeomorphism between the 1-sphere and a semi-open real interval

I need help with a problem that's troubling me. In Lee's "Introduction to Topological Manifolds" I found this exercise: being given the exponential map $\ a:[0,1[\to\mathbb{S}^{1}$, $\ a(s)=e^{2\pi i ...