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

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2
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0answers
16 views

Writing $\mathrm{SO}(2)$ as the zero-set of a function

Here I'm assuming $M_{2 \times 2}(\mathbb{R}) \cong \mathbb{R}^{4}$. The definition of $\mathrm{SO}(2)$ is: $\mathrm{SO}(2)=\{ \ A \in M_{2 \times 2}(\mathbb{R}) \ | \ \det(A)=1 \mathrm{\ and\ ...
5
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1answer
16 views

Is $[N]^\#([N])$ congruent to $w_n(\nu_N)([N])$ mod $2$, where $\nu_N$ is the normal bundle of the embedding of $N$ in $M$?

Let $M$ be a closed, smooth, orientable $2n$-manifold, and let $N$ be a closed, smooth, orientable $n$-submanifold. Let $[N]^\#$ denote the cohomology class (Poincaré) dual to the homology class ...
2
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0answers
40 views

Einstein notation

I'm confused about a specific issue that I have with the Einstein notation (for tensor fields on manifolds). I want to write the following thing: Let $X$ be a smooth manifold. Choosing local ...
0
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0answers
10 views

Volume Problem in Munkres' analysis on manifolds

I am having trouble with problem (a) of this question. I figured that the volume of $\triangle_1(R)$ is $|(\alpha(a+h, b)-\alpha(a, b))\times (\alpha(a+h, b+k)-\alpha(a, b))|$ but don't know how I ...
0
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0answers
17 views

Proof of relation between normal of a surface and principle curvatures of surface.

If F(x,y,z) is a scalar function. Then how to prove that, $$\nabla . n = K_1 +K_2$$ where n is normal to surface of constant $F$ given as $$n=\frac{\nabla F}{|\nabla F|}$$ $K_1$ and $K_2$ are ...
15
votes
1answer
443 views

Can we generalize the regular value theorem even beyond the Ehresmann's theorem?

The formulation is complicated, but the answer may be some clever usage of the partition of unity, because locally the answer is given by the regular value theorem and the whole problem is to glue it ...
2
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0answers
29 views

Poincaré duality isomorphism maps cohomology to homology here?

Let $M = M^n$ and $A = A^p$ be compact oriented manifolds with smooth embedding $i: M \to A$. Let $k = p - n$. Does the Poincaré duality isomorphism$$\bigcap \mu_A: H^k(A) \to H_n(A)$$map the ...
2
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1answer
30 views

Explicit procedure for integrating densities (twisted $n$-forms)?

I'm aware of several slightly different (yet equivalent) definitions for the density bundle of a non-orientable manifold. Unfortunately, I've been unable to make sense of integration in any of them. ...
0
votes
1answer
29 views

Proof that the special linear group $\mathrm{SL}(n,\mathbb{R})$ is a smooth manifold

This is my definition of a smooth manifold I am supposed to work with: Let $\mathcal{M} \subseteq \mathbb{R}^{n}$. The set $\mathcal{M}$ is a $k$-dimensional smooth submanifold of $\mathbb{R}^{n}$ ...
0
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1answer
20 views

How can I calculate this integral of a differential form in a surface?

I'm trying to integrate the 2-form $\omega = A(y) dx \wedge dy - dx \wedge dz + B(z)dz \wedge dy $ in the set $R_f=\{(x,y,z),\quad z=f(x,y)\quad x^2+y^2 \neq 1 \}$ with $f$ a differentiable function ...
-1
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0answers
12 views

Normal bundle and global function [on hold]

Suppose $M \subseteq \Bbb R^m$ is a closed embedded submanifold. If $M$ admits a global defining function, show that its normal bundle is trivial. Conversely, if $M$ has trivial normal bundle, show ...
4
votes
2answers
327 views

Why do differential forms and integrands have different transformation behaviours under diffeomorphisms?

Let $f$ be a diffeomorphism, say from $\mathbb R^n$ to $\mathbb R^n$ , such as the transition map between two coordinate charts on a differentiable manifold. A differential $n$-form (or rather its ...
5
votes
3answers
91 views

Generalized Gauss map, giving rise to second fundamental form

I know that the tangent bundle of $G_n(\mathbb{R}^{n+k})$ is isomorphic to $\text{Hom}(\gamma^n(\mathbb{R}^{n+k}), \gamma^\perp)$, where $\gamma^\perp$ denotes the orthogonal complement of ...
1
vote
1answer
22 views

Proving $\mathrm{GL}(n,\mathbb{R})$ is a smooth manifold

Consider the set $\mathrm{GL}(n,\mathbb{R}) = \{ \ A \in M_{n \times n}(\mathbb{R}) \ | \ \mathrm{det}(A) \neq 0 \ \}$. I'm trying to show that this is smooth submanifold of $\mathbb{R}^{n^{2}} \cong ...
4
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0answers
43 views

Can the concept of orientability be applied to more general spaces?

Today a friend asked me if the Moebius strip with one segment identified to a point is orientable or not. The first thing I replied is that it is not a manifold so you can't define orientability in ...
0
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2answers
28 views

Does the forgetful functor from smooth manifolds to sets preserve colimits?

It is easily shown that the forgetful functor $F: \mathbf{Man} \to \mathbf{Set}$ preserves limits ($F$ is representable), but does it preserve colimits? It certainly preserves all examples of ...
1
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1answer
42 views

How can I calculate the integral $\int_M F^* \omega$?

I got stuck in the following problem. Let $M$ be the manifold defined by the equation $x^2+y^2+z^4=1$ and $F: M \to S^2$ defined as $F(x,y,z)=(x,y,z^2)$. I have to calculate the integral $\int_M F^* ...
0
votes
0answers
15 views

How can I understand that this mapping preserves the orientation of the boundary of this manifold?

Let $M$ be the cylinder of radius 1 (with $z$ between 1 and -1) and $f: M \to M$ the application defined as $f(\cos(\theta), \sin(\theta), t) = (\cos(4\theta), \sin(4\theta), -t)$. I want to give an ...
2
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1answer
14 views

Expression of a given vector field for the stereographic projection of the sphere

I have got stuck trying to solve the following problem. Let $X=-zx \frac{\partial}{\partial x} -zy \frac{\partial}{\partial y} + (1-z^2) \frac{\partial}{\partial z}$ be a vector field in ...
3
votes
0answers
54 views
+50

Codimension 1 Embedding into \mathbb{R}^{n+1}

I am trying to determine which homotopy types can be realized by n-manifolds that have codimension one embeddings into $\mathbb{R}^{n+1}$. Suppose I have $X^n \subset \mathbb{R}^{n+1}$ and a embedded ...
1
vote
1answer
38 views

Calculating Volume of surface of unit sphere

I am trying to understand the proof for $w_n = 2\pi^{n/2}/\Gamma(n/2)$ where $w_n$ is the volume of the surface $S_n$ of the n-dimensional unit sphere $K_n$. There is stated that $Vol(K_n) = ...
0
votes
0answers
13 views

Grassmannian Non-Convex

The Grassmannian manifold $Gr(r,V)$ defines the set of $r$-dimensional linear subspaces of the vector space $V$. My question is, in general, what is the simplest way to see that $Gr(r,V)$ is a ...
1
vote
2answers
27 views

Rank of Jacobian Matrix for the Stereographic Projection

With the definition $S^{n} = \{\ \mathbf{x} \in \mathbb{R}^{n+1}\ | \ ||\mathbf{x}|| = 1\ \}$, and the function $\ f:\mathbb{R}^{n} \to S^{n} \setminus \{ (0,...,0,1) \}$ defined by: $f(\mathbf{u}) ...
5
votes
2answers
94 views

Smooth manifold $M$ is completely determined by the ring $F$.

For any smooth manifold $M$, the collection $F = C^\infty(M, \mathbb{R})$ of smooth real valued functions on $M$ can be made into a ring, and every point $x \in M$ determines a ring homomorphism $F ...
9
votes
1answer
175 views

Is It Always Possible to Cross a Surface Exactly Once?

Yesterday, in my physics class, the following question arose: Is there a closed surface embedded in $\mathbb R^3$ dividing space into two connected components such that all paths from one ...
0
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0answers
43 views

Compact 1-manifolds

I want to find all compact 1-manifolds. I think these will be lines, with fixed endpoints, and also the case where the endpoints meet. Hence I think all compact 1-manifolds are homeomorphic to the ...
6
votes
3answers
181 views

For $n$ even, antipodal map of $S^n$ is homotopic to reflection and has degree $-1$?

How do I see that for $n$ even, the antipodal map of $S^n$ is homotopic to the reflection$$r(x_1, \dots, x_{n+1}) = (-x_1, x_2, \dots, x_{n+1}),$$and therefore has degree $-1$? Thanks for your time. ...
8
votes
2answers
67 views

Diffeomorphism between $\mathbb{P}^n$ and the submanifold of $\mathbb{R}^{(n+1)^2}$ consisting of certain matrices?

Let $\mathbb{P}^n$ denote the set of all lines through the origin in the coordinates space $\mathbb{R}^{n+1}$. Define a function$$q: \mathbb{R}^{n+1} - \{0\} \to \mathbb{P}^n$$ by $q(x) = \mathbb{R}x ...
8
votes
2answers
302 views

On the smooth structure of $\mathbb{R}P^n$ in Milnor's book on characteristic classes.

This question concerns problem 1-B in the book of Milnor and Stasheff part a. They first define the set $F:=\{f:\mathbb{R}P^n\rightarrow\mathbb{R} \mid \text{$f\circ q$ is smooth}\}$ where ...
2
votes
1answer
33 views

Embedding of classical Lie groups

This is somehow very natural question so I'm sure that the answer should be well known: Whitney theorem states that each (say paracompact) $n$-dimensional manifold could be embedded in ...
2
votes
2answers
159 views

The singular homology and cohomology of manifolds vanishes in high dimensions

Let $M$ be an $n$-manifold. It seems that there are two results that the $p$-th singular homology and cohomology of $M$ are zero if $p>n$. But I can not find them in my books of algebraic ...
3
votes
0answers
38 views

Correspondence defines embedding of $G_n(\mathbb{R}^m)$ into $G_{n+1}(\mathbb{R}^1 \oplus \mathbb{R}^m) = G_{n+1}(\mathbb{R}^{m+1})$? [closed]

Does the correspondence$$X \overset{f}{\to} \mathbb{R}^1 \oplus X$$defines an embedding of the Grassmann manifold $G_n(\mathbb{R}^m)$ into $$G_{n+1}(\mathbb{R}^1 \oplus \mathbb{R}^m) = ...
2
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0answers
39 views

Manifolds or Complex Analysis for Algebraic Geometry? [closed]

I'm an undergraduate and I have one year left to take some courses at the graduate level to prepare myself for graduate school. I go to a quarter school (U. Washington) so I only have time to take two ...
0
votes
0answers
35 views

$C^k$-maps between manifolds is a sheaf?

I know that the functor from the category of open subsets of a manifold $M$ to the Sets, taking an open set $U$ and associating to it the collection of $C^k$ maps to $\mathbb{R}$ is a sheaf. My ...
20
votes
3answers
3k views

Under what conditions the quotient space of a manifold is a manifold?

There are many operations we can do with topological spaces that when we apply to topological manifolds gives us back topological manifolds. The disjoint union and the product are examples of that. ...
2
votes
0answers
24 views

Wedge product of $k$-forms

I'm studying smooth manifolds with Lee's book. He defines a $k$-form on a manifold $M$ as a section $M \to \Lambda^k M$ (where $\Lambda^k M = \bigsqcup_{p\in M} \Lambda^k T_pM$ is the smooth vector ...
-2
votes
0answers
18 views

How is vector, dual vector, etc defined in Matrix Lie Group Manifold?

How is vector, dual vector, etc defined in Matrix Lie Group Manifold? Are the coefficients matrices and the (dual)basis matrices as well?For example, the Maurer–Cartan form can be written as ...
0
votes
1answer
39 views

Is $M=\{(x,y,z)\in \mathbb{R}^3: x^3+y^3+z^3=1, z=xy\}$ a manifold of class $C^{\infty}$?

Let $M=\{(x,y,z)\in \mathbb{R}^3: x^3+y^3+z^3=1, z=xy\}$. Is $M$ a manifold of class $C^{\infty}$? I need find a atlas $\{(U_i,\varphi_i)\}_{i\in I}$ with $U_i$ open sets and $\varphi$ ...
0
votes
1answer
43 views

How does the solution of a differential equation on a manifold yield a map?

In: "A solution $x^μ(λ)$ is a map from $\mathbb{R} → M$": Why is $x^μ(λ)$ considered a map and why does it go from $\mathbb{R} → M$? I can't seem to illustrate this in my mind. In:"If the manifold ...
1
vote
1answer
25 views

Completeness of “weighted” shortest path metric

I am trying to see when this type of metric is complete: Let $A$ be the set of $C^{1}$ paths in $U \in \mathbb{R}^{n}$. For any $x,y$ define $$\rho(x,y) = \inf_{\gamma \in A; \gamma(0) = x, \gamma(1) ...
0
votes
2answers
19 views

Brouwer degree extension Lemma

Let $M$ and $N$ be oriented $n$-dimensional manifolds without boundary an also $M$ is compact and $N$ connected. Suppose that $M$ is the boundary of a compact oriented manifold $X$ and that $M$ is ...
4
votes
1answer
88 views

Grassmannian, symmetric, idempotent matrices of trace $n$?

How do I see that $G_n(\mathbb{R}^m)$ is diffeomorphic to the smooth manifold consisting of all $m \times m$ symmetric, idempotent matrices of trace $n$?
2
votes
2answers
83 views

Are $\mathbb{C}^2$ and $\mathbb{C}^2/(x,y)\sim(y,x)$ homeomorphic?

Let $A$ be the set of monic quadratics over $\mathbb C$ and let $B$ be the set of unordered pairs over $\mathbb C$ where possibly the two elements of the pair may be the same. Then the map which takes ...
2
votes
0answers
22 views

Connection on $T\mathbb{R}^n$

Let $\nabla$ be a connection on the tangent bundle $T\mathbb{R}^n$. Now, I need to show that there exist smooth function $C_i: \mathbb{R}^n\to \mathbb{R}^n\times \mathbb{R}^n$, $i=1,\dots ,n$ such ...
0
votes
0answers
18 views

Direct Sum Decomposition and Multiplicity of Eigenvalue Zero

I am currently reading a paper which states the following: Let be given a function $f(x):\mathbb{R}^n \rightarrow\mathbb{R}^n$ with its zero set $\mathcal{V}(f)$. If for every $a\in\mathcal{V}(f)$ ...
1
vote
0answers
24 views

Proving smooth map between smooth manifolds is constant based on push forward being zero

I have just me this problem in my class on smooth manifolds from Lee's introduction to smooth manifolds, from the chapter on the tangent bundle stating the following: Let M, N be smooth manifolds, ...
-2
votes
1answer
38 views

A function $\phi$ between two manifolds of class $C^\infty$ is constant if $d\phi$ [closed]

Let $M$ and $N$ two manifolds of $C^{\infty}$, $M$ connected, and $\phi:M\to N$ also of class $C^{\infty}$ so that in all point $m\in M$ the function $d\phi(m):M_m\to N_{\phi(m)}$ between the ...
0
votes
0answers
40 views

Brouwer Degree is locally constant

I'm reading Milnor's book "Topology from the differential viewpoint" and I'm stuck at this point: Let $M$ and $N$ be oriented n-dimensional manifolds without boundary and let $$f: M \longrightarrow ...
1
vote
1answer
48 views

Stokes theorem for Cuboid

I need to proof stokes theorem $\int_Qd\omega=\int_{\partial Q}\omega\;$ for a 2-form and $Q\subset \mathbb R^3 \;$a cuboid. Since $\omega \;$ is a two form it can be written as $$\omega ...
4
votes
2answers
62 views

If $\mathbb{RP}^n$ can be immersed in $\mathbb{R}^{n+1}$, how do I see that $n$ must be of the form $2^r - 1$ or $2^r - 2$?

If $\mathbb{RP}^n$ can be immersed in $\mathbb{R}^{n+1}$, how do I see that $n$ must be of the form $2^r - 1$ or $2^r - 2$? For starters, I know that if the $n$-dimensional $M$ can be immersed in ...