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

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23 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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11 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
35 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
54 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
28 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 ...
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0answers
25 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 ...
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1answer
18 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 ...
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0answers
25 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 ...
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1answer
32 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
33 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 = ...
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4answers
769 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 ...
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1answer
46 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 ...
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1answer
66 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 ...
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1answer
41 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 ...
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1answer
23 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 ...
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0answers
31 views

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

In picture below ,how to get the equality in red box?
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1answer
34 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 ...
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1answer
94 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 ...
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1answer
44 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
24 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
42 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 ...
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1answer
175 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 $S^3$. ...
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0answers
16 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 ...
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2answers
31 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 $A$ ...
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1answer
22 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 ...
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1answer
32 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
40 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 $D_{c(t)}$, ...
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1answer
56 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 $\mathbb R^n$.
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2answers
24 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 ...
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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 ...
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27 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?
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2answers
35 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 ...
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0answers
38 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. $$g(X_1+X_2,Y_1+Y_2) = g_1(X_1,Y_1)+g_2(X_2,Y_2),$$ ...
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1answer
27 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 ...
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2answers
24 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 ...
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0answers
49 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 ...
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0answers
8 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 ...
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0answers
45 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 ...
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2answers
19 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 ...
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0answers
18 views

Star of cell homeomorphic to $B_{n}$ in regular CW decompositions of manifold?

Let $(X,X_{i})$ be a regular CW decomposition of a $n$-manifold and let $e$ be a cell of $(X,X_{i})$. We define the star of $e$ as the union of $e$ with all cells $e'$ with $e$ on the boundary and ...
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0answers
12 views

Show that $X = \{(x,-1):x\in \mathbb{R}\} \cup \{ ( \cos ( \theta ), \sin ( \theta ): 0 \leq \theta < 2\pi\}$ is NOT a topological 1-manifold

I'm having difficulty proving this. I have a feeling it has something to do with the point $(0,-1)$, and the property that topological manifolds are required to be locally $E^n$ (like an open set ...
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1answer
43 views

Map isotopic to identity is orientation preserving

Let $M$ be an $n$-dimensional orientable and compact smooth manifold and $f:M\to M$ be a smooth map isotopic to the identity map. Is it true that $f$ is orientation preserving?
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1answer
35 views

Compact manifold, regular value

I am still trying to convince myself about a fact which I've seen in Milnor's book ''Topology from the differentiable viewpoint'': Let M and N be manifolds of the same dimension. If M is a compact ...
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1answer
33 views

Dual Spaces vs Dual Bases

I'm trying to wrap my head around differentiable manifolds and tensors. I partially worked through a question which asked me to use the metric tensor and the line element in spherical polar ...
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1answer
16 views

A homeomorphism of B^n fixing the boundary?

I am trying to construct an automorphism $$\phi:\mathbb{\overline B}^n\to\mathbb{\overline B}^n$$ such that $\phi(0) = \alpha\hat x_1$, and $\phi|_{\partial\mathbb{\overline B}^n} =$ Id. I thought ...
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0answers
34 views

compute compactly supported cohomology of a space

Given space $M=R^2/0$ ($R^2$ excluding origin), how to compute compactly supported de Rham cohomology of M, in notation, $H^*(c(M))$? I am thinking of raising a vector v in $R^2$ with |v| at most ...
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0answers
28 views

From orthogonal group to orthonormal frames

Define the orthonormal frame of a space $R^n$ to be a set of vectors ($b_1$, $b_2$, ..., $b_n$) if these vectors together form an orthonormal basis of $R^n$, and denote this frame to be $F_n$. ...
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1answer
51 views

How to prove the given sets are equal?

I did not understand the highlighted part(in red). How is the equality obtained? One inclusion is clear namely, the left hand side is a subset of the right hand side. Why is this an open subset ...
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0answers
18 views

Is a regular point of a smooth function also a regular point for the restriction to a submanifold?

Suppose I have a smooth map $F \colon M \rightarrow N$ and an embedded submanifold $S \subset M$. Suppose that the differential of $F$ is non-singular at a point $z \in M$. If $z$ also happens to lie ...
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0answers
27 views

Proving $O(1,1)$ is diffeomorphic to the disjoint union of 4 copies of the real line.

I'm trying to prove that the pseudo-orthogonal group $O(1,1)$ is diffeomorphic to the disjoint union of four copies of $\mathbb{R}$. Firstly I tried to solve the equation $AI_{p,q}A^T = I_{p,q}$ where ...