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

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1answer
47 views

Product of smooth maps from $M \to \mathbb{R}$ is smooth

In Lee's Introduction to Smooth Manifolds there is a exercise to proof that if $f: M \to \mathbb{R}$ and $g: M \to \mathbb{R}$ are smooth, so is $fg$. The question I'm having is the following: By ...
2
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1answer
55 views

embeddings of projective spaces into Euclidean spaces

Let $\mathbb{R}P^n$, $\mathbb{C}P^n$, $\mathbb{H}P^n$ be the real, complex, quaternionic projective spaces resp. I want to find all $n$ such that $\mathbb{R}P^n$ can be embedded into $\mathbb{R}^{n+1}...
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2answers
23 views

Existence a path of a smooth manifold

Given a continuous differentiable functio $F:\mathbb{R}^n\mapsto \mathbb{R}^m$ with $n>m$. Define $$ {\cal M}=\{x\in\mathbb{R}^n: F(x)=0\} $$ and let $x_0\in{\cal M}$ such that the Jacobian of ...
0
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1answer
33 views

Example of germs not involving series

note: moved from mathoverflow, as off topic. I'm currently reading this book: http://www.springer.com/us/book/9781441973993 And when speaking about germs of functions the only example provided of two ...
6
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2answers
158 views

(Anti-) Holomorphic significance?

What are holomorphic and anti-holomorphic components? Why don't we call them complex components and their conjugates? What is holomorphic coordinate transformation?
7
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1answer
53 views

Is the union of an increasing sequence of topological copies of $\mathbb{R}^n$ homeomorphic to $\mathbb{R}^n$?

Let $M$ be an $n$-dimensional topological manifold, and let $(U_k)_{k \in \mathbb{N}}$ be an increasing sequence of open sets $U_k \subset M$ such that for each $k \in \mathbb{N}$, $U_k$ is ...
2
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1answer
36 views

Is the set of points where two functions agree a submanifold?

Let $M, N$ be smooth manifolds, $f : M \to N$ a smooth submersion and $g : M \to N$ a smooth function. Is it true that $R = \{x \in M\ |\ f(x) = g(x)\}$ is an embedded submanifold of $M$? How about if ...
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2answers
136 views

$M$ not orientable implying results about $H_{n-1}(M, \mathbb{Z})$, $H_n(M, \mathbb{Z}_q)$?

Let $M$ be a compact connected $n$-manifold without boundary, where $n \ge 2$. How do I see that if $M$ is not orientable, then the torsion subgroup of $H_{n-1}(M, \mathbb{Z})$ is cyclic of order $2$ ...
0
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1answer
21 views

Show that $c(t)_{c(t)}$ and the tangent vector to $c$ at $t$ is perpendicular.

Let $c(t)=(c_1(t),...,c_n(t)):[0,1] \to \Bbb R^n$ be a differentiable curve in $\Bbb R^n$ such that $|c(t)|=1$ $\forall t \in [0,1]$. Define tangent vector $v$ of $c$ at $t$ as $c_*((e_1)_t)=((c_1)'(t)...
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1answer
46 views

When is the inverse image of continuous function a submanifold

Let $f : R^n → R$ be a continuous function. Let $c ∈ Im f$. When is $f ^{−1}(c)$ a manifold? I know that if $f$ is a smooth function then $f ^{−1}(c)$ is a manifold if $c$ is a regular value. But ...
6
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2answers
139 views

$M$ orientable implies $H_{n-1}(M, \mathbb{Z})$ is free Abelian group. [closed]

Let $M$ be a compact connected $n$-manifold without boundary, where $n \ge 2$. How do I see that if $M$ is orientable, then $H_{n-1}(M, \mathbb{Z})$ is a free Abelian group?
2
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0answers
33 views

Understanding the definition of submanifold

I have defined a submanifold as: $M \subset \mathbb{R^n}$ is called a $k$-dim submanifold of $\mathbb{R^n}$ if $\forall x \in M$ the following condition holds: $\exists U \subset \mathbb{R^n}$ open, ...
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0answers
13 views

How to prove that the half cone doesn't have a tangent space at its vertex?

I don't know how to prove that the half cone (including its vertex) $\lbrace (x,y,z): x^2+y^2=z^2 , z\geq 0 \rbrace$ does not have a tangent space at its vertex (0,0,0).
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0answers
58 views

Barycentric subdivision of regular CW decomposition is a combinatorial manifold?

Suppose $X$ is a PL manifold (with boundary) and let $(X,X_{i})$ be a regular CW complex. Is the barycentric subdivision of $(X,X_{i})$ a combinatorial manifold? Answer given in comments. Definition: ...
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0answers
88 views

Duality between tangent and cotangent bundles

Given a smooth manifold $M$, the cotangent bundle $T^*M$ is dual to the tangent bundle $TM$ "fiberwise", i.e. $\forall x\in M$, $T^*_x(M)=(T_x(M))^*$. Now, if the manifold is a vector space, then the ...
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0answers
32 views

How to characterize the dimension of a manifold using homology?

This might be a trivial question but I'm a physicist, not a mathematician. For me, the n dimensional euclidean space is n dimensional as a vector space. I have heard however that there more intrinsic ...
1
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0answers
40 views

Lifting of triangulation

In "Complex Analysis 2: Riemann Surfaces, Several Complex Variables, Abelian Functions, Higher Modular Functions" and many other books is described a lifting of triangulations for branched covers ...
0
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1answer
24 views

Tangent vectors in $T_p\partial M$

I know that if $M$ is a smooth $n$-dimensional manifold with boundary, then $\partial M$ is a smooth $(n-1)$-dimensional manifold. So for $p\in\partial M$, we have $T_p\partial M\cong\mathbb{R}^{n-1}\...
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0answers
29 views

Quintic equation and number of lines on the quintic

I heard a talk where the speaker said that the solution to the equation $x_1^5 +x_2^5 +x_3^5 +x_4^5 +x_5^5 = 0$ is a six-dimensional (Calabi-Yau) manifold. Then he went on to define five curves of ...
3
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1answer
48 views

Tangent space of open set [duplicate]

If U is a open subset in $\mathbb{R}^n$ and p is a point in U, then tangent space of U at point p is the whole of $\mathbb{R}^n$. I am having difficulty understanding why this is true. Why is the ...
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0answers
36 views

What it means to “put together all the maps” here?

I'm reading Spivak's Mechanics book and he says the following when talking about Hamiltonian Mechanics Given a Lagrangian $L : TM\to \mathbb{R}$, at each point $a\in M$ the restriction $L_a = L|...
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4answers
955 views

Is every manifold a metric space?

I'm trying to learn some topology as a hobby, and my understanding is that all manifolds are examples of topological spaces. Similarly, all metric spaces are also examples of topological spaces. I ...
1
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1answer
61 views

The cone is not a regular submanifold of $\mathbb{R}^3$

I am not very familiar to differentiable manifolds, so I would appreciate some hints or reasonings about why the cone $$ M = \{(x,y,z)\in\mathbb R^3:x^2+y^2-z^2 = 0, z\geq0\} $$ is not a regular ...
4
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1answer
73 views

Meaning of different manifold structures

I would like to prove that if $$ M = \{(x,y)\in\mathbb R^2: y^2 - 4x^2(1-x^2) = 0\} $$ and $$ P:(0,2\pi)\to M, \quad \theta \mapsto (\sin \theta,\sin 2\theta), $$ $$ Q:(-\pi,\pi)\to M, \quad \theta \...
0
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1answer
45 views

About differential structures, manifolds

I am dealing with the following problem on Differential manifolds and I don't know how to solve it. If anyone could please help me, I would be very thankful. Tanks in advance. Let $\lbrace (x,y)\in \...
0
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1answer
27 views

Partial derivatives on manifolds in terms of local charts

Let $\phi=(u^1,\cdots, u^n)$ be a coordinate system in manifold $M$ at $p$. If $f \in c^{\infty}(M)$, we define $$\frac{\partial f}{\partial u^i} (p) = \frac{\partial(f \circ \phi ^{-1})}{\partial x^...
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0answers
36 views

Contraction of $(2k,2l)$tensor

In picture below ,how to get the equality in red box?
2
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1answer
36 views

Can we consider compact sets of Riemannian manifolds as ones of closed Rimanninan manifolds?

Let $(M,g)$ be a $C^\infty$ Riemannian manifold of $n$ dimensional and suppose $\emptyset\neq K \stackrel{\mathrm{compact}}{\subset} M$. Then are there any neiborhood $\Omega\stackrel{\mathrm{open}}\...
3
votes
1answer
100 views

Non-existence of embedded incompressible surfaces

I want to prove the following assumption: Let $g,h$ be natural numbers with $g > h$ and let $S_g$ be the closed, orientable surface of genus $g$. Then, there is no (smooth) map $f: S_g \to S_h \...
1
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1answer
46 views

A beginner's question of Riemannian Geometry.

In picture below ,I don't know why $\Phi^{-1}(F)=(F(\phi^i,e_j))_{i,j=1}^n$
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1answer
30 views

Example of fibre bundle is locally product but not globally

When I read the below picture ,I can't make a example for claim of red box.
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1answer
48 views

Can someone help me understand the Euclidean metric?

A Euclidean metric is defined as: $g_{euclid} = g_{ij} = \delta_{ij} dx^i \times dx^j = dx^1dx^1 + \ldots + dx^ndx^n$ Can someone explain the following: why do we use $dx^i$ instead of $x^i$ which ...
7
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1answer
276 views

Visualization of SU(3)

I am trying to visualize the $SU(3)$ group used in quantum field theory. I have a (reasonably) good understanding of $SU(2)$ as the double cover of $SO(3)$ and also that this is homeomorphic to $S^3$. ...
0
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0answers
41 views

Understanding the differentiable structures on Grassmann manifold

I am reading in the book Differential Analysis on Complex Manifolds by Raymond. I have a trouble in understanding the differentiable structures on Grassamann space. I uploaded the picture of the page ...
0
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2answers
81 views

Topological Boundary vs Manifold Boundary

Let $A$ be the open unit disc in $\mathbb{R}^2$ and $B$ be the closed unit disc in $\mathbb{R}^2$. The toplogical boundary of $A$ and $B$ is $S^1$. This I understand. The manifold boundary of $A$ ...
0
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1answer
32 views

The composition of a smooth map with a linear isomorphism is smooth

I'm reading Jeffrey Lee's Manifolds and Differential Geometry section on manifolds with boundary. In page 50, given an $n$-dimensional manifold $M$, he builds a smooth atlas for $\partial M$ in the ...
2
votes
1answer
36 views

Definition of symmetric product in Milnor's paper

I am currently reading Milnor's paper which discusses the group action on spheres without fixed point. At the second page of the paper, he denotes $$M^n*M^n$$ to be a symmetric product of a manifold. ...
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0answers
48 views

Tangent Spaces & Definition of Differentiation

What is meant by definition when we talk about the $c:(-\epsilon,\epsilon)\rightarrow M$ is that an interval on the curve? In differential geometry what is the difference between $D_{c(t)}$, $Df_{...
1
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1answer
80 views

Construction of Grassmann manifolds

Is there a way to construct the Grassmann manifold via block matrices? For example the upper triangular matrices stabilize the (coordinate) basis of $\mathbb R^n$.
1
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2answers
25 views

Is it possible to represent (up to a $λ^3$-null set) a 2-dimensional submanifold of $R^3$ as the graph of a $C^1$-function $f:U⊆R^2\to R$?

Let $M$ be a two-dimensional submanifold of $\mathbb R^3$. Is $M$ (globally) representable as the graph of a continuous differentiable function $f:U\subseteq\mathbb R^2\to\mathbb R$ in the sense, that ...
0
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1answer
28 views

Can we represent the interior of a polygon as the graph of a continuously differentiable function?

Let $P$ be a polygon in $\mathbb R^3$. Can we find an open set $U\subseteq\mathbb R^2$ and a continuously differentiable function $f:U\to\mathbb R$ such that $$\text{int}(P)=\left\{\left(x,f(x)\right):...
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0answers
30 views

Submersions and transversality in terms of orthogonality and factorization systems?

Are the notions of submersion and transversal maps somehow a special case of the categorical notion of orthogonal morphisms and factorization systems?
2
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2answers
38 views

If $M$ is a submanifold of $\mathbb R^3$ and the normal space $N$ on $M$ at $p$ is one-dimensional, can we choose an unique “outer” normal from $N$?

Let $M$ be a two-dimensional submanifold of $\mathbb R^3$. Then, the normal space $N_p(M)$ on $M$ at $p\in M$ is one-dimensional. So, there are only two unit normal vectors $n_1$ and $n_2$ on $M$ at $...
2
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1answer
78 views

Curvature on product Riemannian manifolds

I am working on the following problem from Lee's Riemannian Manifolds: Suppose $g = g_1 \oplus g_2$ is a product metric on $M_1 \times M_2$ (i.e. $$g(X_1+X_2,Y_1+Y_2) = g_1(X_1,Y_1)+g_2(X_2,Y_2),$$ ...
0
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1answer
30 views

Why is a Vanishing Vector Field $\Rightarrow$ Discontinuous Frame?

I'm reading Tu's Introduction to Manifolds section on oreintablility. On an orientable manifold, if any vector field vanishes at a point $p$ (e.g. $S^2$), then the global frame $(X_1,X_2)$ that ...
3
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2answers
30 views

Number of Orientations of Disconnected Manifold

This seems like a stupid question, but the number of orientations of a smooth manifold with $n$ maximal connected components would be $2^n$, right? Since each connected component $U\subset M$ is open $...
0
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0answers
57 views

Lagrangian in pseudo-Riemannian manifold and geodesics

I'm trying to solve the following problem without success. Let $V$ be a smooth function on a pseudo-Riemannian manifold $(M, g)$, which is either bounded from above or from below. Show that there ...
0
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0answers
13 views

About the number of minimum parametrizations of a $1$-smooth manifold compact w/ boundary in $\mathbb{R}^{3}$

Let $C$ be a $1$-dimensional compact differentiable manifold with boundary in $\mathbb{R}^{3}$. In easy examples, it looks like we can always parametrize such a manifold with only two charts: usually ...
5
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0answers
51 views

Maximizing the pairwise Frobenuis distance between M othrogonal matrices

I want to maximize the pairwise Frobenius distance between $M$ orthogonal matrices. That is, I'm looking for $Q_{i}, i = 1, 2, ... M$ such that \begin{equation*} \begin{aligned} & \underset{ 1 \...
1
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2answers
28 views

Union of the unit circle and the line $y=-1$ is not a topological $1$-manifold.

I'm reviewing past papers for a topology exam, and I can't answer this question on topological manifolds: Let $$X= \{(x,-1):x\in \mathbb{R}\}\cup\{(\cos \theta, \sin \theta ): 0 \le \theta \le 2\pi \...