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

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3
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1answer
42 views

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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0answers
17 views

Given a smooth curve denoted by $(f(u),0,(g(u))$, if the rotation of the curve around the $z$-axis is a manifold, is $g$ one-to-one? Is it onto?

Given a smooth curve denoted by $(f(u),0,(g(u))$, if the rotation of the curve around the $z$-axis is a smooth manifold, is $g$ one-to-one? Is it onto? By rotation, I mean $(u,v) \rightarrow (f(u)...
2
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1answer
98 views

Prove: $h \in T_xM \iff \operatorname{dist}(x+\epsilon h,M) = o(\epsilon)$

Prove: $h \in T_xM \iff \operatorname{dist}(x+\epsilon h,M) = o(\epsilon)$ For $M$ a smooth manifold in $\mathbb{R}^n$, $h\in\mathbb{R}^n$, $x \in M$. I know that $T_xM = \operatorname{Im}(Df) = \ker(...
0
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1answer
20 views

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 ...
1
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1answer
32 views

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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0answers
25 views

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 ...
1
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1answer
50 views

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, $X',Y'=Df(X),Df(...
10
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1answer
89 views

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 ...
28
votes
2answers
985 views

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 ...
6
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1answer
46 views

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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0answers
46 views

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 ...
0
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0answers
23 views

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 ...
2
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1answer
31 views

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 ...
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1answer
56 views

Showing that diffeomorphisms between manifolds preserves orientability

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 ...
2
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1answer
78 views

The quotient of $\Bbb R^3$ by a finite group.

Let $\Gamma$ be a finite subgroup of $SO(3)$ acting on $\Bbb R^3$. What sort of space do we get by taking the quotient $\Bbb R^3/\Gamma$? Is that a manifold? The group $\Gamma$ is compact since it is ...
0
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0answers
36 views

Dual basis cotangent space

I have been given the unitary sphere in the Euclidean space. $$F(\theta, \phi) =(\sin\theta \cos\phi, \sin\theta \sin\phi,\cos\theta)$$ I'm asked to show that the dual base of $E_1=F_*(\partial \phi)...
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0answers
29 views

Degree of Gauss map coincides with Euler characteristic

Let $M^n \subset \mathbb{R}^{n+1}$ be a compact hypersurface, oriented with the smooth normal vector field $N(X) \perp T_xM$. Let $G: M^n \to S^n$ be the corresponding Gauss map. Does it follow that $\...
2
votes
1answer
67 views

Not sure about my proof that orthogonal matrices are a manifold in ${\rm Mat}_{n \times n}(\mathbb{R})$

Not sure about my proof that orthogonal matrices are a manifold in ${\rm Mat}_{n \times n}(\mathbb{R})$ I know that the manifold is the zero set of the function $f(A) = AA^T - I_n$. The thing I don'...
2
votes
1answer
61 views

Is the intersection of $x^2 + y^2 + z^2 = 1$ and $x = \frac{1}{2}$ a manifold in $\mathbb{R}^3$?

Is the intersection of $x^2 + y^2 + z^2 = 1$ and $x = \frac{1}{2}$ a manifold in $\mathbb{R}^3$? I think that it is, because it can be parameterized by $f(x) = (\frac{1}{2},\sqrt{\frac{3}{4}} \cos x, ...
2
votes
1answer
38 views

Definition of Integral Morse Homology

I am reading through "Morse Homology and Floer Homology" by Audin and Damian and I am confused about the definition of the differential in integral Morse homology. Let $V$ be a compact manifold, $f:...
3
votes
1answer
39 views

Is “Let $M$ is a connected n-manifold, then: $M$ is $R$-orientable iff $H_n(M;R)\cong R$” true?

I heard that every topological $n$-manifold $M$ is $\mathbb{F}_2$-orientable, but then for $M=\mathbb{R}^2$ is must be $H_2(\mathbb{R}^2;\mathbb{F}_2)\neq 0$? In lecture we had the lemma: Let $M$ is ...
2
votes
2answers
89 views

Show that $SL(n, \mathbb{R})$ is a $(n^2 -1)$ smooth submanifold of $M(n,\mathbb{R})$

I need to show for $n=3$ that $SL(n,\mathbb{R})=\{A \in M(n, \mathbb{R}) : detA=1 \}$ is a $(n^2 -1)$ dimensional smooth submanifold of the vector space $M(n,\mathbb{R})$ of all real $n \times n$ ...
0
votes
1answer
39 views

Show there exists a unique map $g$ such that $g \circ f_{2} = h$

I was wondering if somebody could give me some help on this question. Any hints etc. would be greatly appreciated. Let $0 < b < a$. Define a smooth map $h: \mathbb{R}^2 \rightarrow \mathbb{R}^3$...
0
votes
1answer
55 views

Find a diffeomorphism between $SO(3)$ and $\mathbb{R}P^3$

Find a diffeomorphism between $SO(3)$ and $\mathbb{R}P^3$ $SO(3) =$ {${A \in M_3(\mathbb{R}) : A^TA = I, \det(A) = 1}$}, which is the special orthogonal group. And $\mathbb{R}P^3$ is the real ...
3
votes
1answer
51 views

$H_d^k(n\text{-torus})$ isomorphic to $\Lambda^k(\mathbb{R}^n)$.

Let $M^n = \mathbb{R}^n/\mathbb{Z}^n$ be the $n$-torus. Show that $H_d^k(M^n)$ is isomorphic to $\Lambda^k(\mathbb{R}^n)$. My thoughts on the problem are as follows. The map $G_{ty}(x) = x + ty$, $...
0
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1answer
39 views

Is the set $x^2-y^2 = 0, z > y \geq0 $ a smooth manifold in $\mathbb{R}^3$?

Is the set $x^2-y^2 = 0, z > y \geq0 $ a smooth manifold in $\mathbb{R}^3$? I think that the answer is no, but I'm not really sure how to prove this as I'm having trouble visualizing how it looks. ...
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0answers
31 views

Necessary condition for the intersection of two submanifolds to be a submanifold

Let $X$ be an $n$ dimensional manifod. How could I prove that for arbitrary submanifolds $M,N$ of dimension $n-m,n-k$, if $\forall x\in M\cap N$ $dim( T_xM\cap T_xN)=n-m-k$ then $M\cap N$ is a ...
2
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0answers
16 views

Evaluate $\int_Mxdy\wedge dz$ where $M$ is the torus formed by the circle of radius $1$ in the $xz$ plane centered at $(2,0,)$ rotated around $y$ axis

Evaluate $\int_Mxdy\wedge dz$ where $M$ is the torus obtained by rotating the circle $(x-2)^2+z^2=1$ around the $y$ axis. I've parameterized $M$ using $\alpha:(0,2\pi)\times (0,2\pi)\rightarrow M$ ...
0
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0answers
15 views

Using Fubini's and Fundamental Theorem, explain why $\int_Sdx\wedge dy$=$\int_{\partial S}xdy$ where $S=\{(x,y,z)\in\mathbb R^3|x^2+y^2+z^2=1,z>0\}$

This is sort of an odd question because it's asking to justify generalized Stokes' Theorem using only "Fubini's" (i.e. equality of iterated integrals) and Fundamental Theorem of Calculus (i.e. 1-...
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0answers
46 views

Describe an atlas of smoothly related charts for the Special Orthogonal Group $SO(3)$

The Special Orthogonal Group $SO(3) =$ {${A \in M_3(\mathbb{R}) : A^TA = I, det(A) = 1}$} I have successfully shown that $SO(3)$ is a manifold, but I am having a difficult time explicitly finding a ...
0
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2answers
46 views

Parametrize $\{(x,y,z)\in\mathbb{R}^3\mid x+y^2+z^2=1, x\geq0\}$

Let $M =\{(x,y,z)\in\mathbb{R}^3\mid x+y^2+z^2=1,x\geq0\}$. It seems to me that this manifold is a "cone" since we have $y^2+z^2=1-x$ for $x\in[0,1]$ which, geometrically, is a circle in the $yz$ ...
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0answers
13 views

Find the volume of h(P) in terms of volume of P

Where h:$\mathbb R^n \rightarrow R^n$ is the function h(x) = $\lambda x$ and P is a k-dimensional parallelopiped in $\mathbb R^n$ Attempt at the solution : Let $x_1,....,x_k$ be vectors in $R^n$ ...
0
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1answer
47 views

Question on the definition of outward normal vector from Spivak, Calculus on Manifolds

The following definition of the outward unit normal at the boundary of a manifold $M \subseteq \mathbb R^n$ is taken from Spivak, Calculus on manifolds (page 119). If $M$ is a $k$-dimensional ...
2
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1answer
41 views

The defintion of orientation of a manifold from Spivak, Calculus on Manifolds

In Spivak Calculus on Manifolds the author uses a definition of orientation of a manifold which I do not understand, and which I do not found elsewhere. I cite: It is often necessary to choose an ...
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0answers
35 views

Counterexample of the fundamental theorem for hypersurfaces in the Euclidean space $\mathbb{R}^{n+1}$ in global sense

Are there any example of a closed Riemannian $n$-manifold $(M,g)$ and a symmetric bilinear form $A=h_{ij}dx^i\otimes dx^j\in\Gamma(T^\ast M\otimes T^\ast M)$ satisfying Gauss' equation \begin{eqnarray}...
2
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1answer
38 views

Question on how differential form as defined for subsets of $\mathbb R^n$ and integration on them in Spivak, Calculus on Manifolds

If $V$ is a vector space, denote by $\Lambda^k(V)$ the space of alternating multilinear maps from $V^k$ to $\mathbb R$, i.e. the space of alternating $k$-tensors. Also for a point $p \in \mathbb R^n$ ...
3
votes
1answer
39 views

Locally disk-preserving charts?

This is slightly vague as I've not yet come to terms with what I'm actually looking for. On $S^2$ we may choose charts (stereographic projection) such that the image of a disk (i.e. all points ...
0
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0answers
19 views

Chain rule over manifolds

Let $X,Y$ and $Z$ be three manifolds, and let $F:X\to Y,G:Y\to Z$ be smooth functions. Fix a point $x\in X$. Prove that $$D(G\circ F)\mid_{x}=DG\mid_{F(x)}\circ DF\mid_{x}$$ By using coordinate ...
4
votes
1answer
105 views

If $\delta c = 0$, does it follow that $d\xi = 0$?

Let $\mathcal{U} = \{\mathcal{U}_\alpha\}_{\alpha = \infty}^\infty$ be a locally finite open covering of the manifold $M^n$, with smooth functions $\lambda_\alpha$, compactly supported in $U_\alpha$. ...
3
votes
1answer
70 views

De Rahm cohomology of a sphere, help with proof

I am working through Guillemin and Pollack's proof that the de Rahm cohomology of the sphere is $H^p(\mathbf{S}^k) = \mathbf{R}$ for $p = 0$ and $p = k$ and $H^p(\mathbf{S}^k) = 0$ otherwise. Here, $...
1
vote
1answer
51 views

If a manifold has a submanifold, then the local space is a cartesian product or splits in some other way?

The following definitions are taken from Marsden et al. Manifolds, Tensor Analysis, and Applications. Definition 1: Let $S$ be a set. A chart on $S$ is a bijection $\varphi$ from a subset $U$ of $...
0
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1answer
41 views

A question on an exercise to show that unit sphere could not be covered by a single chart

The following is an exercise from Marsden et al, Manifolds, Tensor Analysis, and Applications in the first chapter on manifolds. First let me cite three essential definitions: Definition 1: Let $S$...
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2answers
39 views

Munkres Analysis on Manifolds Differentiation Question

Below is a problem from Munkre's Analysis on Manifolds book. I'm unsure of how to approach this; it seems to me to apply the defintion of the derivitative, but I cannot seem to get that to work out. ...
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0answers
33 views

Description of Frechet derivatives and the implicit function theorem

[QUESTION] Let $(S^n,\bar{g})$ be the unit sphere and $h$ be another Riemannian metric on $S^n$, $0<\alpha<1$. $M^{2+\alpha}(S^n):=\left\{F:S^n\stackrel{C^{2,\alpha}}\to S^n\right\}$. For $F:(S^...
0
votes
2answers
37 views

Cartesian Product of Compact Set and Non-Compact Set is Non-Compact

Theorem: Let $A$ be a compact set and $B$ be a non-compact set. Then $A\times B$ is non-compact. I know that if $B$ is non-compact, then there exists an open cover $O$ of $B$ that does not have a ...
4
votes
0answers
37 views

If $M = \partial W$, with $W$ parallelizable, then an embedding $\iota : M \to S^{n+k}$ extends to an embedding $W \to D^{n+k+1}$

Suppose $M$ is an $n$-dimensional $s$-parallelizable manifold which is the boundary of the parallelizable compact manifold $W$. It is claimed in Milnor & Kervaire's Groups of Homotopy Spheres ...
1
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1answer
37 views

A maybe naive question about the definition of manifolds and the common analogy with real-world atlases

Maybe this question is a little bit naive, but there might be a good explanation. I am reading about manifolds, manifolds of dimension $d$ are defined, for example to be subsets $M \subseteq \mathbb R^...
4
votes
2answers
57 views

signature of the topological manifold $M= \mathbb{C}P^6\times \mathbb{C}P^6$ (using homology, cohomology)

I want to prove that the signature $\operatorname{sig}(M)$ of the topological manifold $M= \mathbb{C}P^6\times \mathbb{C}P^6$ is nonzero. First of all, $M$ is a compact 24-dimensional manifold ...
1
vote
1answer
46 views

Two figures, one is not a manifold, but the other is a manifold, but they look both as suffering from the same deficiencies for not being a manifold

I am looking video lectures of F. Schuller about space-time geometry, in particular about manifolds. In it, right at the beginning after introducing manifolds he gives a non-example, i.e. he says that ...
1
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0answers
28 views

How does an almost complex structure on a manifold induce an orientation?

I have read that given a smooth even dimensional manifold $M$ with an almost complex structure $J$, then $M$ is orientable and there is a canonical choice of orientation. Why is this the case? How ...