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

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30 views

If $\gamma : J \mapsto M$ is a smooth curve in a smooth manifold M, then $\gamma'(t) \neq 0$ $\forall t \in J$ iff $d\gamma$ is injective.

If $\gamma : J \mapsto M$ is a smooth curve in a smooth manifold M, then $\gamma'(t) \neq 0$ $\forall t \in J$ iff $d\gamma$ is injective. Here $J$ is just an open interval of $\mathbb{R}$ I'm just ...
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1answer
41 views

Existence of a nonzero vector to form

Let $ f: \mathbb {R}^m\times \mathbb {R}^m \rightarrow \mathbb {R}^m $ an alternate form of grade two. If $ m $ is odd, prove that there exists $ v\neq 0 $ such that $ f (u, v) = 0 $, for all $ u \in ...
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1answer
53 views

How prove that $\mathbb{CP}^2$ is compact? [closed]

How prove that $\mathbb{CP}^2$ is a compact manifold.
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0answers
32 views

An $n-1$ dimensional surface has $n$ dimensional measure $0$.

How does one show this? I was thinking that on an $(n-1)$ dimensional surface there a local homeomorphism to $\mathbb{R}^{n-1}$, which can be canonically embedded into $\mathbb{R}^n$, and it seems ...
5
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1answer
90 views

Volume of neighborhood of the curve

Let $\gamma:[0,1] \to \mathbb R^n$ be a smooth closed curve, such that $\gamma(0)=\gamma(1), \gamma'(0)=\gamma'(1), |\gamma'(a)|=1$ and let $B_r=\{x| \exists t, |\gamma(t)-x|<r\}$. How can i show ...
3
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1answer
43 views

1-form on $S^n$ with non-degenerate differential.

How it can be solved? Whether there is a $1$-differential form $w$ on $S^n$, such that $dw$ is non-degenerate on $T_aS^n$ for each $a \in S^n$?
3
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1answer
143 views

Euler characteristic is equal to self-intersection number of zero-section?

As I recall (from Guillemin and Pollack "Differential Topology") the Euler characteristic of a (for my purposes, compact and oriented) smooth manifold X is defined as $\chi(X)=I(\Delta,\Delta)$, ...
1
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1answer
36 views

How to show that the metric in the tangent space is independent from the chart you take?

I want to prove that for vectors $v_1,v_2 \in T_aM$ the euclidean length and distance is independent from the chart we are using, where $M$ is a submanifold in some $\mathbb{R}^n$ My problem is that ...
2
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1answer
77 views

Derivative of antipodal map between $n$-spheres

Let $S^{n-1}\subseteq \mathbb{R}^n$ denote the $(n-1)$-sphere $x_1^2+\ldots+x_n^2=1$. Let $f:S^{n-1}\rightarrow S^{n-1}$ be the map $f(x_1,\ldots,x_n)=(-x_1,\ldots,-x_n)$. What is the derivative of ...
1
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1answer
32 views

Kernel of matrix with identity as submatrix

Let $f:\mathbb{R}^n\rightarrow\mathbb{R}^k$ be a $C^\infty$ map and let $X=\text{graph}f$, i.e. $$X=\{(x,y)\in\mathbb{R}^n\times\mathbb{R}^k\mid y=f(x)\}.$$ What is the tangent space to $X$ at ...
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1answer
28 views

Computing tangent space for quadric

What is the tangent space to the quadric $x_1^2+x_2^2+\ldots+x_{n-1}^2=x_n^2$ at the point $p=(1,0,\ldots,0,1)$? The definition of a tangent space that I know is based on the fact that we have a ...
3
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1answer
47 views

Quadric in $n$ dimensions is a manifold?

I can show that the sphere $x_1^2+x_2^2+\ldots+x_n^2=1$ is an $(n-1)$-dimensional manifold by considering the map $f(x_1,\ldots,x_n)=x_1^2+\ldots+x_n^2$, and noticing that $1$ is a regular value of ...
5
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0answers
39 views

Domain invariance for smooth functions

The domain invariance theorem states that for an open set $U\subset \mathbb{R}^n$ and a continuous and injective mapping $f:U\to \mathbb{R}^n,$ the image $f(U)\subset \mathbb{R}^n$ is open. I've read ...
3
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2answers
31 views

System of two equations form a manifold

Show that the set of solutions of the system of equations $$x_1^2+\ldots+x_n^2=1$$ and $$x_1+\ldots+x_n=0$$ is an $(n-2)$-dimensional submanifold of $\mathbb{R}^n$. I want to take ...
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0answers
28 views

what is definition of para kenmotsu manifolds?

Thank you for your attention. Frist I study on para kenmotsu manifolds anad I study 3 article baut I don't finded definition of it. why have we In para hermitian manifold whay h(JX,JY)=- h(X,Y) but in ...
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1answer
31 views

Another exercise from Fleming's Functions of Several Variables.

I'm using Flemming's book Function of Several Variables. In it, the author defines Manifolds like this: Let $1\le r\lt n,\ q\ge1$. A nonempty set $M \subset \mathbb{R}^n$ is a manifold of dimension ...
0
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1answer
43 views

Intersection of a manifold with open set

I'm using Flemming's book Function of Several Variables. In it, the author defines Manifolds like this: Let $1\le r\lt n,\ q\ge1$. A nonempty set $M \subset \mathbb{R}^n$ is a manifold of dimension ...
1
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1answer
81 views

How to prove that a vector bundle is trivial iff there are n global sections that form a basis on each fiber?

I can prove the only if part. My attempt to prove if part is the following: Given $n$ global sections $s_1, s_2, ..., s_n$ of a vector bundle $E$ on a smooth manifold $M$ such that they form a basis ...
3
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1answer
50 views

Does this Manifold exist?

The excercise is the following: Give an example or disprove: There is at least one m-dimensional manifold that is compact in some $\mathbb{R}^n$ such that one chart is sufficient to get the whole ...
7
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2answers
72 views

Detecting compactness from the ring of smooth functions

Given a smooth manifold $M$, is there some ring-theoretic property (preferably not mentioning $M$) such that $C^{\infty}(M)$ has this property if and only if $M$ is compact?
0
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1answer
75 views

Compute the geodesic curvature of any sphere on a sphere.

Compute the geodesic curvature of any sphere on a sphere. Again there exists its answer, but not understandable for me. Please explain it explicitly. Thank you so much. (If required, i can post ...
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2answers
53 views

Curve is a submanifold

In our class today we said today that an 1-times continuously differentiable immersion $\phi:[0,1] \rightarrow \mathbb{R}^n$ is a submanifold of dimension 1 if it is closed such that $\phi(0)=\phi(1)$ ...
2
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1answer
44 views

Is there a connected compact manifold $M$ of dimension 4 such that $\pi_1 (M) = \mathbb{Z} * (\mathbb{Z} \oplus \mathbb{Z}_3)$?

Is there a connected compact manifold $M$ of dimension 4 such that $\pi_1 (M) = \mathbb{Z} * (\mathbb{Z} \oplus \mathbb{Z}_3)$? I had this question in a test yesterday. I think that the answer is no, ...
5
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1answer
148 views

CW complexes and manifolds

What is the strongest known theorem (if any) that classify which manifolds can be built as CW complexes? Thank you guys. This is just a question I thought of during class, obviously not homework... ...
1
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1answer
61 views

Second fundamental form question.

Honestly, I dont have any idea for that question I posted. Please can someone help me solving the question. Thank you.
0
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1answer
78 views

Transition map for Möbius band in differential geometry.

Calculate the transition map $\phi$ between the two surface patches for the möbius band. These two surface patches are the following $U=\{(t,\theta) \ | -1/2\lt t\lt 1/2,\ \ 0\lt \theta \lt ...
4
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0answers
63 views

An imcomprehensible proof on Arnold's Lectures on PDE, contact diffeomorphism

In Lectures on Partial Differential Equations, Arnold gives a theorem in the chapter Huygens' Principle in the Theory of Wave Propagation: Theorem 1 (the theory of support functions). The manifold ...
3
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1answer
72 views

Vector on a manifold

I can't fully understand the step that is made in this diff. geometry textbook. It says: For $c:(a,b)$ $\rightarrow$ $M$ and function $f: M$ $\rightarrow$ $\mathbb{R}$ where $(a,b)$ is an open ...
3
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2answers
60 views

Riemannian Manifolds with $n(n+1)/2$ dimensional symmetry group

Given a $n$-dimensional connected Riemannian manifold $(M,g)$, its symmetry group $G$ can be considered as a subbundle of orthonormal frame bundle of $M$ (which I call $F_OM$), yielding: $$\dim G\le ...
3
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1answer
98 views

Non-vanishing vector fields on non-compact manifolds

In several papers the following result is invoked: Theorem. Every connected, non-compact, smooth manifold $M$ carries non-vanishing smooth vector fields $v$. (we are assuming $M$ is $2$nd countable ...
10
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1answer
170 views

Applications of TQFTs beyond physics

I'm giving a talk at a postgrad seminar on the topic of topological quantum field theories (TQFTs) with a mixed audience of pure and applied mathematicians. As such, I'd like to be able to offer some ...
2
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1answer
55 views

Constructing submanifolds. Did I understand this right?

I just want to know whether I understand the construction of a submanifold in some $\mathbb{R}^n$ properly. Please correct everything that you think could be wrong. As far as I know so far, it is ...
1
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1answer
114 views

Inner product on tangent space and metric tensor

In our class we talked about integrating on submanifolds and as a short side remark our teacher told us that by knowing the metric tensor, it is possible to define an inner product on a tangent space ...
2
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0answers
81 views

Definition of a tangent space

Today we defined a tangent space similar to the description here: enter link description here My problem is the following: Why do we need to refer to charts in this case? I mean, would it not be ...
3
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2answers
133 views

A non orientable closed surface cannot be embedded into $\mathbb{R}^3$

Can someone please remind me how this goes? Here's the idea of proof I'm trying to recall: let $S$ be a closed surface (connected, compact, without boundary) embedded in $\mathbb{R}^3$. Then one can ...
1
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1answer
59 views

Lie algebra homomorphism and action on a manifold

In Introduction to smooth manifolds Lee says on page 527: If $\mathfrak{g}$ is an arbitrary finite-dimensional Lie algebra, any Lie algebra homomorphism ...
2
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0answers
34 views

Analytic/Smooth/Continuous maps between a manifold and itself

Let us suppose that $M_{\omega}$ is a connected real-analytic manifold of dimension $n$. Then there is an associated smooth structure, $\mathcal{C}^r$ structure ($r$ non-negative integer) on it. Let ...
3
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1answer
51 views

smooth lie group action

Let $\theta:G\times M\to M$ be a smooth left action of a Lie group $G$ on the manifold $M$. Suppose $G$ is compact and $M$ is Hausdorff. Let $K$ be a compact set in $M$. Is it true that $G_K:=\{g\in ...
5
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1answer
208 views

Are there p-adic manifolds?

Is there anything resembling a manifold on the field of p-adic or complex p-adic fields? If so is there a connection to algebraic geometry as rich as in the reals?
3
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1answer
43 views

write the trasition map $\phi$ between $\sigma_1$ and $\sigma_2$. Verify $\det( J(\phi))$ and find $T_p(S)$.

Sphere $$S=\{(x,y,z) \mid x^2+y^2+z^2=R^2\}$$ $$ \sigma_1(u,v)=(u,v, \sqrt{1-u^2-v^2}) \\ \sigma_2(u,v)=(\tilde u, \sqrt{1-\tilde{u}^2 -\tilde{v}^2}, \tilde v) $$ I guess $\{\sigma_1, \sigma_2\}$ ...
2
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1answer
35 views

Definition of a one-connected manifold?

Perhaps the question is self-explanatory. The context is Kleiner's Inv. Math. paper An isoperimetric comparison theorem, where the statement of the main theorem begins with "Let $M$ be a complete ...
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1answer
29 views

Significance of rank of frechet derivative in definition of manifold?

In studying manifolds, the stipulation that the derivative be full rank is confusing to me on an intuitive level. Can anyone please explain how I should think about this intuitively? What does it mean ...
2
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1answer
80 views

Is $\mathbb Z$ a submanifold of $\mathbb R$?

Is $\mathbb Z$ a sub-manifold of $\mathbb R$? If yes, what kind of sub-manifold it is? Thanks
0
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1answer
85 views

Orientability of manifold

How to prove the following theorem (or where the detailed proof can be seen, since it isn't proved in lectures I've attended): Two basis of tangent space of manifold have same orientation if matrix ...
0
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1answer
69 views

Integral manifold

I'm solving the following: Let $D$ be a distribution on $\mathbb{R}^3/\{(0,y,z):y,z\in\mathbb{R}\}$ with basis vector fields $X=z\frac{\partial}{\partial x}-x\frac{\partial}{\partial z}$ ...
0
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1answer
48 views

Diffeotopy and connectedness on manifolds

Let $M, N$ manifolds without bundary and $f: M\to N$ and $g: M\to N$ embeddings. We say that a differentiable map $h: [0,1]\times M\to N$ is an isotopy between $f$ and $g$ if each of the maps ...
0
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0answers
28 views

Express the change of coordinate matrix in terms of partial derivative of the transition map $\phi$ Where $\tilde{\sigma} =\sigma \circ \phi$

Let $\sigma :U \to W\cap S$ and $\tilde{\sigma}: \tilde{U} \to \tilde W \cap S$ Be two surface patches around $p\in W\cap \tilde W$ $T_p(S) $ be tangent plane of S at p. $T_p(S)=span\{\sigma_u, ...
5
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1answer
46 views

Why are there always pairwise intersections in a Heegaard splitting?

Let $M=A\cup B$ be a Heegaard splitting, such that $\{\alpha_i\}_{i=1}^g$ is a set of boundaries for meridian disks of $A$, and $\{\beta_i\}_{i=1}^g$ is a set of boundaries for meridian disks of $B$ ...
2
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1answer
174 views

Tensors constructed out of metric other than the Riemann curvature tensor

Let $(M,g_{ab})$ be a Riemannian (or Pseudo-Riemannian) manifold and let us define a tensor field as 'something' that transforms in an appropriate way under coordinate transformations. (This is how ...
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1answer
97 views

The double cone is not a surface.

My question is that A double cone ( also named as "circular cone") is not a surface. I know its reason. But I cannot show this mathematically. Suppose $\sigma : U \to S\cap W$ Is a surface ...