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

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80 views

compactification of $\Bbb R^n$

I want to show that $n$ dimensional real projection space is compactification of $\Bbb R^n$, how can I do that? I can show PR^n is compact, and R^n is localy compact, but I have problem to show R^n ...
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26 views

Showing that a set is a smooth submanifold

How do I show that the solution set to the equations $x^3+y^3+z^3=1$ and $z=xy$ is a smooth manifold? Regular value theorem doesn't apply here and I don't know how to construct an atlas for it.
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1answer
42 views

Derivative of map $f: S^n \to \mathbb{R}P^n$ is an isomorphism

I'm trying to show that the map $f: S^n \to \mathbb{R}P^n$ given by sending a unit vector $x$ in $S^n \subset \mathbb{R}^{n+1}$ to the line spanned by $x$ in $\mathbb{R}P^n$ has injective derivative. ...
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1answer
44 views

Why is a metric?

I have a question about tensors and metrics: Let $M=\{(t,x,y,z)\in \mathbb{R}^4: t>-1 \}$ and let $g=(1+t)dtdx+dy^2+dz^2$ Show that g is a metric on $M$. I did the next, I have the basis $\{ ...
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1answer
30 views

On computing the Differential of a Smooth Map

In class, we proved Jacobi's formula for the differential of the determinant using the following formula for the differential of a smooth map $F$ between manifolds $M$ and $N$ $$D_A F(B) = ...
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1answer
52 views

Why is $\phi^* g = g$ a PDE for a pseudo-Riemannian metric $g$ on a manifold?

Given a (locally trivial) bundle $\pi: E \to M$ a PDE of order $k$ is usually defined to be a submanifold of the jet-bundel $J^k(E)$. Now assume $E = M \times M$ and $\pi$ is the projection on the ...
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1answer
42 views

What is the skew-symmetric part of the covariant derivative of a one-form?

This is a followup question to here. Let $E \to M$ be a vector bundle with connection $D := \nabla$. Extend $D$ to $E^*$ and $\text{Hom}(E, E)$. Let $E = TM$ here, and suppose that the torsion is ...
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0answers
69 views

Which homology groups of a closed orientable 6-manifold can be isomorphic to $\mathbb{Z}^3$?

List all $i$ for which there is a closed orientable $6$-manifold $M$ with $H_i(M) =\mathbb{Z}\oplus\mathbb{Z}\oplus\mathbb{Z}$ I am working on an old exam problem and this one stumped me. ...
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1answer
77 views

Why is the image of the implicit function in the implicit function theorem not open?

We have a continuously differentiable function $f$ from $\mathbb{R}^{n+m}$ to $\mathbb{R}^n$, and we find a continuously differentiable function $g$ which maps points from $\mathbb{R}^m$ into ...
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1answer
34 views

Difference between Grassmann and Projective space?

I recently read the first few chapters of Pyramid Algorithms: A Dynamic Programming Approach to Curves and Surfaces for Geometric Modeling and I am confused about ...
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1answer
57 views

Question about connections on the dual bundle.

Let $E \to M$ be a vector bundle with connection $\nabla$. Extend $\nabla$ to $E^*$ and $E^* \otimes E$ in the regular fashion. Is $\text{Id} \in E^* \otimes E$ necessarily parallel?
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19 views

Quadrature over a (smooth, compact, convex, etc.) Riemannian manifold

Problem setting Consider three points on the surface of the earth (which I want to assume to be a perfect ellipsoid here) that are pairwise sufficiently close for unique geodesics to be found between ...
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0answers
32 views

When and where to check the formal definition of a manifold

In most texts on GR we are first introduced to a formal and rigorous definition of a manifold. We then learn the point that in GR "any coordinate system" might be used for the 4D spacetime metric. ...
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1answer
25 views

Reference request for stochastic processes on manifolds

I'm looking for some references on stochastic processes on manifolds. The more introductory the better. Thanks.
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1answer
48 views

Every $\mathcal{C}^1$ manifold can be made smooth?

I heard of a theorem saying that each $\mathcal{C}^k$-manifold with $k\geq 1$ can be made into a smooth manifold, i.e. $\mathcal{C}^{\infty}$ (by restriction of the atlas). However, I cannot find ...
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1answer
29 views

Lemma characterizing second fundamental form, do not understand step

Consider an excerpt of a lemma and part of its proof from a Riemannian geometry textbook. Lemma. The second fundamental form is independent of the extensions of $X$ and $Y$; bilinear ...
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1answer
42 views

Quotient by a discrete subgroup of a Lie group

I was reading Fulton Harris' Representation theory, A first course, where I came across the following: Let $H$ be a Lie group and $T$ be a discrete subgroup of its center $Z(H)$. Then there exists ...
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1answer
30 views

Understanding Hempel's proof of uniqueness of cube with handles

In Hempel's 3-Manifolds book, Theorem 2.2 says that if $P$ and $Q$ are two cubes, both with $n$ handles, and both are orientable, then they are homeomorphic. He defines a cube with handles as a ...
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1answer
60 views

Are the collections {rings}, {smooth manifolds} larger or smaller than {ab. groups}, {top. manifolds}?

Intuitively, the collection of smooth manifolds feels smaller than that of topological manifolds: they are not just locally nice continuous objects, but even smooth. Similarly, when one finds that an ...
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1answer
25 views

Is every convex cone a manifold?

Let $C \subseteq \mathbb{R}^n\setminus \{0 \}$ be a connected convex cone*. Question: Is $C$ always a topological manifold (perhaps with boundary)? A smooth one? Does anything change if we do not ...
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1answer
32 views

Vector bundle vs Total Space

On page 59 in Lee's "An Introduction to Smooth Manifolds" the author writes, "Let $E$ be a smooth vector bundle over a smooth manifold $M$, with projection $\pi:E\to M$." I thought the vector bundle ...
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0answers
30 views

Does anything obstruct Mostow-Prasad rigidity for orbifolds?

If we phrase the Mostow-Prasad rigidity theorem algebraically, it goes like this (let $\mathcal{H}^n$ be a model for hyperbolic $n$-space). For $n>2$: if ...
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2answers
30 views

Choice of chart and altas

Manifold is second countable space. Should charts in altas also be countable? I don't think it has to be. But some how the second countable condition enforces me to think like that..
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34 views

Is $T^2 \# \mathbb RP^2\cong \mathbb RP^2\# \mathbb RP^2\# \mathbb RP^2$?

I was reading a little about how to imagine the projective plane and I have some weird intuition that says $T^2 \# \mathbb RP^2\cong \mathbb RP^2\# \mathbb RP^2\# \mathbb RP^2$. Is this true, and if ...
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2answers
51 views

Representable homology classes on smooth manifolds

Let $X$ be a closed (compact without boundary) smooth manifold. We can consider its singular homology $H_*(X,\mathbb{Z})$. Let $H_{k}(X,\mathbb{Z})$ be the $k$-th singular homology group of $X$ and ...
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20 views

Poincaré-Hopf Index Theorem - Intuitive explanation

I heard about an interesting theorem. Poincaré-Hopf Index Theorem : If $\vec{v}$ is a smooth vector filed on the compact, oriented manifold $X$ with only finitely many zeros, then the global ...
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1answer
25 views

Tensor field acting on vector fields and covector fields gives a function?

If I have a $(p,q)$ tensor field that acts on $p$ covector fields and $q$ vector fields then does $T\left(X_1,\dots,X_p,Y_1,\dots,Y_q\right)$ return a function $f$ defined on the manifold by ...
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1answer
57 views

The classifying space of open covers of a manifold

Let $M$ be a manifold of dimension $d$ and let $\mathsf{Disk}_{/M}$ be the category of open subsets of $M$ that are diffeomorphic to $\mathbb{R}^d$ with morphisms given by inclusions. Let $\mathrm{B} ...
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1answer
23 views

Calculating the second fundamental form of surfaces

I am asked to prove that for a surface in $\mathbb{R}^3$ with local coordinates in a chart, u,v, the coefficients of the second fundamental form can be calculated as follows: eg.: (first entry in II) ...
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1answer
37 views

Integral of Differential 1-form

Let $\omega$ be the closed $1$-form $\omega = \frac{xdy - ydx}{x^2 + y^2}$ and let $S$ be the unit circle and let $C = \{(x,y) : (x-3)^2 +y^2 = 1\}$. I'm trying to find $\int _S \omega$ and $\int ...
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1answer
24 views

Integrate 2-Form over surface

Problem: Calculate $\int_S dx \wedge dy + dy \wedge dz$, where $S$ is the surface given by $S = \{(x,y,z) : x = z^2 +y^2 -1, x < 0\}$. Wikipedia says: Let $$ \omega=f_{z}\, \mathrm dx \wedge ...
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0answers
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Sum two nearest function of two class are the nearest function of the sum class

Suppose $x,\mu:[0,1]\rightarrow \mathbb{R^2}$ two smooth function and $\Gamma = \{\gamma : [0, 1] \rightarrow [0, 1]| \gamma (0) = 0, \gamma (1) = 1, \gamma$ is a diffeomorphism $\}$. Here $\Gamma$ ...
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1answer
31 views

Differential of the multiplication and inverse maps on a Lie group

I'm trying to solve the following two problems: Let $G$ be a Lie group with multiplication map $\mu\colon G\times G\to G$, and let $\ell_a\colon G\to G$ and $r_b\colon G\to G$ be left and right ...
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1answer
32 views

Why does $(X_a,Y_b)(f\circ \mu(a,b)) = X_a(f\circ R_b(a)) + Y_b(f\circ L_a(b))$?

On the first page of these notes: if $\mu\colon G\times G\to G$ is the multiplication map on a Lie group $G$, then given a point $(a,b)\in G\times G$ and letting $R_b$ and $L_a$ denote ...
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1answer
60 views

Apparent violation of fundamental theorem of ODEs, how to resolve?

Consider, in the $(x, y)$-plane, the family of curves given by $y = (x - c)^3$, for the various possible values of the number $c$. Denote by $v$ the unit vector field everywhere tangent to this family ...
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0answers
26 views

$M=M_1\cup M_2$ is not necessarily a manifold, when $M_1\cap M_2=\emptyset$, $M_i$ a manifold

While $M_1\cap M_2=\emptyset$, $M_i$ a manifold, show $M=M_1\cup M_2$ is not necessarily a manifold. Another question, prior to this one, was to show the union is a manifold, where the conditions ...
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0answers
5 views

Vector bundle morphisms $T(I\times I)\longrightarrow A$?

Let $I:=[0, 1]$ be the unit interval in $\mathbb R$ and $\pi:A\longrightarrow M$ a vector bundle. Is there a nice characterization of the vector bundle morphisms $T(I\times I)\longrightarrow A$? ...
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2answers
33 views

Diffeomorphism group of product manifold

For a given differentiable manifold $M$, the diffeomorphism group $\mathrm{Diff}\left( M \right)$ of $M$ is the group of all $C^\infty$ diffeomorphisms of $M$ to itself. Consider a product manifold of ...
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1answer
25 views

Whats the surface area of the surface $0 \leq z, (x-1)^2 + y^2 \leq 1$?

Whats the surface area of the manifold $0 \leq z, (x-1)^2 + y^2 \leq 1$? The surface is the intersection of the sphere $x^2 + y^2 + z^2 = 4$ and a cylinder centered at $(1,0,0)$. I'm just not sure how ...
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0answers
29 views

Linearity in quotient space

Let $\mathcal{C}$ be the space of all parametric curves $x:[0,1]\rightarrow \mathbb{R}^2$. Also let $\mathcal{C}$ is a linear manifold in the sense that $x_1,x_2\in \mathcal{C}$ implies that ...
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1answer
29 views

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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1answer
97 views

Potential proof for the Slice-Ribbon conjecture (may be wrong).

Let $f:(D^2,S^1)\to(D^4,S^3)$ be a smooth embedding (so called a slice disk), and we set $M:=f(D^2)$. Then, is the restriction map $C^{\infty}(D^4)\to C^{\infty}(M)$ open map with relative to $C^2$ ...
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0answers
20 views

How to construct free subgroup of SO(3) using Baire's Theorem?

I'm looking to demonstrate that there exists a free group on two generators $ G\subseteq SO(3) $ (this is for homework, so hints are preferred to complete answers I've turned it in now, so full ...
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1answer
40 views

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 ...
3
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2answers
57 views

Is there a locally compact, locally connected, Hausdorff and second countable space that is “nowhere locally Euclidean”?

When I study topological manifold, I think some property of manifolds are so important that they can "almost characterize" manifolds. But I know a topological manifold is not easily to be ...
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32 views

Prove: The pre-image of a null-set on a manifold in $\mathbb{R}^k$ is a null-set

Prove: The pre-image of a null-set on a manifold in $\mathbb{R}^k$ is a null-set. Given a $k$ dimensional manifold, $M$, and a mapping $r: U \rightarrow M$, and a null-set $E \subset M$, prove that ...
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1answer
37 views

Why are are integrals of functions in coordinates on manifolds not invariant under coordinate transformations?

I'm reading the book Introduction to Smooth Manifolds. And there is a question that confuse me on page 202. Can anyone tell me why it would change under coordinate transformations graphically? Thank ...
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0answers
33 views

Differential of a map including a manifold

Let $f\in C^{k}(M,\mathbb R)$ with $M$ is a $m$-Manifold and $d_xf:T_xM\to\mathbb R$ is the surjectiv differential. Let $m\lt l$ and $L:\mathbb R^l\to\mathbb R^{m-1}$ be a linear map and ...
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3answers
464 views

What are the “technical troubles” with using a metric space rather than a topological space when defining an abstract manifold? (As in Spivak)

One thing I think is interesting about Spivak's book A Comprehensive Introduction to Differential Geometry is that Spivak uses metric spaces instead of topological spaces when defining an abstract ...
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1answer
39 views

calculate the surface of the manifold in $\Bbb{R}^4$

How to calculate the surface area of the following manifold : $$ x_1^2 + x_2^2 = x_3^2 + x_4^2, 0 \le x_1^2+x_2^2 \le a^2$$ I know I should first describe this manifold as a map or a graph of a ...