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

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106 views

A subset $A$ of a manifold $X$ that is a manifold but not a submanifold of $X$

Let $X$ be a manifold and $A$ a subset of $X$. Is it possible for $A$ to be a manifold without being a submanifold of $X$. Thank you for your help!!
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0answers
33 views

Proving a global property of the connection from a property of local connection matrices

Let $D$ be an $m$-dimensional distribution on an $n$-dimensional manifold $M$. Let $U\subset M$ be an arbitrary open subset such that on $U$ we can define vector fields $e_i$ such that for each $x\in ...
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0answers
50 views

Tensor vector bundle construction

$\newcommand{\p}{\partial}$Let $M$ be a smooth manifold, and define $$T_{r,s} := \bigsqcup_{p \in M} (T_p M)_{r,s} = \bigsqcup_{p \in M} \big( \underbrace{T_p M \otimes \dots \otimes T_p M}_r \otimes ...
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1answer
31 views

Involutive Properties of Space-structures on Smooth Manifolds

I am currently reading Quantum Invariants of Knots and 3-Manifolds by Turaev, and I am having a hard time understanding a statement made on page 120. He is explaining the property of space-structures, ...
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1answer
164 views

Pullback of a form using the Hopf fibration

I embed the 2-sphere and the 3-sphere in $\mathbb{R}^3$ and $\mathbb{R}^4$ respectively. Then denote by $\{x_1,x_2,x_3,x_4\}$ the coordinates on the 3-sphere and $\{y_1,y_2,y_3\}$ on the 2-sphere. So ...
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1answer
148 views

Geodesics and manifold

I apologize if my post is "silly" because I don't know much about riemannian geometry. I know that $M = (0,1)$ (the open unit interval) can be seen as a one-dimensional manifold. Since $M$ is an ...
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1answer
48 views

Lie brackets definition

Let $v,w$ be vector fields on a smooth manifold $M$ (i.e. $v : M \rightarrow TM = \lbrace (p, v_p) : p \in M, v_p \in T_p M \rbrace$). The Lie brackets of $v,w$ are defined as $$ [v,w](f)|_p = ...
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0answers
75 views

Structure equations on the 3-sphere

On the 3-sphere I have found the vector fields $X_1=(-x_2,x_1,-x_4,x_3)$, $X_2=(-x_3,x_4,x_1,-x_2)$, $X_3=(-x_4,-x_3,x_2,x_1)$, in the basis $\left\{\frac{\partial}{\partial ...
5
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0answers
169 views

Fundamental solution to Laplace equation on arbitrary Riemann surfaces

So, I've seen in a few places this method of calculating the heat kernel on a manifold given the kernel of its universal cover, through a so-called 'tiling method' as in section five of this paper ...
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0answers
115 views

Flowing a vector along a vector field $X$ using the pushforward of the flow of $X$

On the three-sphere $S^3$, I'm given three vector fields $X$, $Y$ and $Z$, such that at each point $p\in S^3$, the tangent vectors $X_p$, $Y_p$ and $Z_p$ form an orthogonal basis of the tangent space ...
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3answers
101 views

Embeddings (how to prove them exactly)

For which of the following sets is the statement: '$A$ can be embedded in $B$' true? I can try to decide this intuitively but don't know if I'm right, and surely don't know how to formally prove it. ...
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2answers
88 views

Surface of graph of a function.

Let $G:=\{(x,y,z)\in\mathbb{R}^3:|x|<|z|^2,|y|<|z|,0<z<1\}$ and $f:G\to\mathbb{R},f(x,y,z)=2x+2y+z^3$. Calculate the surface of the graph of f. We recently got introduced to Stokes' ...
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2answers
397 views

How many charts are needed to cover a 2-torus?

Can you please answer this question with explanation ? I just learned about the charts needed to cover 1 and 2 spheres but got confused for the case of torus. It would be great if you guys could help. ...
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1answer
113 views

Tangent vector to a curve on a manifold

If one has a curve $\sigma : (-1,1) \rightarrow M$, where $M$ is a smooth manifold, the tangent vector in $\sigma(0)$ is usually defined as $$ \sigma'(0) (f) = \dfrac{d f \circ \sigma}{dt} \Big|_0,$$ ...
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1answer
43 views

Proving that charts are related

Let $A$ be an atlas on the set $M$ and let $ x: U \to x(U) $ and $ y : V \to y(V )$ be bijections from subsets $U, V \subset M$ to open sets $x(U), y(V ) \subset \mathbb{R^n} $. Show that if the ...
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0answers
98 views

How to prove that the composite function is smooth

Let $f:M \to N$ and $g:N \to K$ be smooth functions, where $M,N$ and $K$ are smooth manifolds. How to prove that the composite function $g \circ f$ is smooth, noting that the Chain Rule only applies ...
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1answer
67 views

How to verify that this is a submanifold

Let $ g: \mathbb{R}^2 \to \mathbb{R}^2 $ , $ g (x, y) = (x^2-y^2, y) $ be a differentiable map. Let $ r $ the line passing through $(1, 0) $ parallel to the $ y-$axis. Prove that $ g^{-1}(r) $ is a ...
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1answer
116 views

Manifold is $2$nd countable iff it has a countable atlas

I am trying to prove that if a smooth manifold has a countable atlas than it has a countable basis. If I have $(U_n, \varphi_n)$ a countable atlas, how can I find a basis of the topology? If I take ...
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2answers
148 views

Recommendation on studying Smooth Manifold & Differential Geometry and related subjects.

My major was physics, but i'm changing my major to mathematics this year. I took 2 years off to study mathematics myself and now i'm going back in this year. Here's the list of books i have studied ...
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1answer
119 views

prove that a map $g: M \to N$ is smooth

Let $A$ be an $n \times n$ orthogonal matrix (that is, the columns of $A$ are perpendicular to one another and have length $1$). Then multiplication by $A$ defines a map $f:\mathbb{R}^n \to ...
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1answer
46 views

Proof of Morse inequalities?

Can you think of a proof of Morse inequalities without using the Morse cohom. $\cong$ sing.cohom or Witten's approach? http://en.wikipedia.org/wiki/Morse_theory#The_Morse_inequalities Any references ...
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1answer
50 views

integration of differential forms on covering space

Let $M_1,M_2$ be $n$-dimensional oriented manifolds. Let $f: M_1\longrightarrow M_2$ be an orientation-preserving diffeomorphism. Then for any $\omega\in \Omega^n_c(M_2)$ we have(page 85 of {Madsen: ...
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1answer
68 views

What is the best (in terms of effectively building understanding) direction from which to approach manifolds?

Thedore Frankel's book The Geometry of Physics presents Manifolds right away in Chapter 1 in the following manner: Introduce the Euclidean space $\mathbb{R}^N$ only as "the most important manifold". ...
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1answer
22 views

Atlases and Transition Maps

What is the difference between open sets and open balls? Definitions of atlases for manifolds do not seem to specify any difference.
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1answer
29 views

Vanishing of 1-form

If $\theta \in \frak{X}^* \mathrm{(M)}$ and $\theta (X) = 0$ $\forall X \in \frak{X} \mathrm{(M)}$ then $\theta = 0$. How do I prove this statement? Consider a manifold $M$ with chart $x^1, \dots, ...
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0answers
64 views

An (extended) semimetric on surfaces

Given a smooth surface $S \subseteq \mathbb{R}^3$, like the surface of sphere, we can define the following extended semimetric $d : S^2 \to [0, \infty]$, where $$ d(x,y) = \inf\{\lVert x - p\rVert + ...
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1answer
48 views

Significance of bump function in the proof.

In the book Semi-Riemannian Geometry with applications to Relativity by Barrett O'Neill, in Chapter 2 (Tensors), he stated that: However, I don't get why he had to use a bump function in the proof. ...
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1answer
101 views

Topological Manifolds & Covers

This problem is from John Lee's "Introduction to Smooth Manifolds" 1-4. Let M be a topological manifold, and let U be an open cover of M . (a) Assuming that each set in U intersects only finitely ...
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1answer
110 views

$V$ vector field, $\omega$ one-form, $V(\omega(V))$=?

(1-forms) Let $X$ be a manifold and $\omega \in \Omega^1(X)$ be a smooth 1-form, and $V, W \in V^{\infty}(X)$ smooth vector fields on $X$. Then, $\omega(V ), \omega(W ) \in C^{\infty}(X)$ are ...
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0answers
107 views

Lie Bracket and vector fields [closed]

Could you please help me solve it? Let X and Y be smooth vector fields on $\mathbb{R}^n$. Suppose that $[X,Y]=0$ (Lie bracket). Show that around each point there exists local diffeomorphism f such ...
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1answer
51 views

Let $f:\mathbb{R^n} \to \mathbb{R^m}$, if $f$ is a linear transformation, prove that $Df(a)=f$.

Let $f:\mathbb{R^n} \to \mathbb{R^m}$, if $f$ is a linear transformation, prove that $Df(a)=f$. My try : By definition of derivative of a function $f:\mathbb{R^n} \to \mathbb{R^m}$ , If I know ...
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1answer
29 views

Is a one-form a derivation on $C^ \infty$?

I know that a vector field is a derivation on $C^ \infty$, meaning that it is R-linear and Leibnizian. Is it the same case for one-forms?
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0answers
46 views

Cone is a submanifold iff it is a vector subspace

Could you tell me how to prove the following? Let $\emptyset \neq M \subset \mathbb{R}^n $ be a cone ($\forall x \in M, t \in \mathbb{R} : tx \in M $ ), $0 \le d \le n.$ Prove that $M$ is a $d$ ...
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1answer
158 views

Geometric interpretation of Supersymmetry

Is there a geometric interpretation of supersymmetry? I.e., if one has a manifold $\mathcal {M} $, and there are $\mathcal {N} $ SUSY generators, then is there a geometric interpretation of the SUSY ...
3
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0answers
78 views

Local dimension of graph embedding

I am trying to find a way to characterize the dimension of the smallest space into which a (neighbourhood of) a graph $\Gamma = (V, E)$ may be embedded. Although in the end my goal is to identify what ...
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1answer
160 views

Lower bound for the size of an atlas

This question came up in a graduate-level class on differential topology I'm currently taking; the instructor couldn't come up with an answer off the top of her head and while I'm very new to the ...
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0answers
62 views

Compactness and Poincare duality

I am reading Appendix B in Fulton's Young Tableaux about Borel-Moore homology. In particular, I'd like to understand why for compact manifolds the Borel-Moore homology groups are isomorphic to ...
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2answers
33 views

Multilinearity of the exterior derivative of a one-form.

I wish to show that the exterior derivative $d \theta$ of a one-form $\theta$ is $\frak{F} \mathrm{(M)}$-multilinear, therefore, a tensor. Let $X, Y, V, W \in \frak{X} \mathrm{(M)}$ and $f, g, h, k ...
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1answer
147 views

Is this an abuse of notation?

Here is a proof says that the differential of Gauss map is self-adjoint. But I seems there is an abuse of notation at (1) in it. Since $dN_p$ is linear, it suffices to verify that $\langle ...
2
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1answer
83 views

How to decide if a given set is a manifold

Could you tell me how to decide if a certain set is a manifold? I know there already is a similar question here, but there we have fairly "visualizable" sets: a hemisphere and a square. What in ...
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1answer
54 views

Vanishing of a covector (1-form) and a vector field

a) A one-form $\theta$ is zero if and only if $\theta X = 0$ for all $X$ in the set $\frak{X} \mathrm{(M)}$ of all smooth vector fields on a manifold $M$. b) A vector field $X$ is zero if and only ...
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1answer
79 views

Path or One-Dimensional Manifold

The terminology of curve, path and one dimensional manifold is always in the textbooks about topology, differential manifold and riemannian geometry. The definition of path and one dimensional ...
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2answers
118 views

Self-contained text on characteristic classes

I am looking for a clear, self-contained text (either a book or lecture notes) that deals with characteristic classes, starting from the very basics (fiber bundle, principal bundle etc.), and ...
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0answers
58 views

Vector fields on homogeneous space $G/H$

I am trying to understand why the vector fields on $G/H$ are maps $X:G\rightarrow \frak{g}/\frak{h}$ satisfying $X(rh)=Ad^{-1}(h)X(r),\,\, h\in H.$ Any hint would be greatly appreciated!
3
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1answer
255 views

Partial derivative of a function on manifold

Bishop and Goldberg define ("Tensor analysis on manifolds") the partial derivative of a smooth function on a manifold $M$ in the following way: $\partial_i f= \frac{\partial ...
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0answers
64 views

Finding the components of the Riemannian tensor given the components of a metric.

I am looking at a manifold of dimension $n$ (And I am considering a local co-ordinates system $x_1,x_2,\ldots x_n$) and the metric defined by the components $g_{ij} = \frac{\delta_{ij}}{x_1^2}$. I'm ...
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1answer
422 views

Tangent space of Grassmannian $Gr_k (\mathbb{R}^n)$

I am trying to show that the tangent space of the Grassmannian $Gr_k (\mathbb{R}^n)$ at $L,$ is naturally/canonically isomorphic to $Hom(L,\mathbb{R}^n/L).$ However, I cannot see even intuitively what ...
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1answer
90 views

Differential operator on a manifold in Geometric Calculus

In the context of Geometric Calculus, as stated in book Clifford Algebra to Geometric Calculus (pag. 142), let $M$ be a differentiable vector manifold, $F$ be a field on $M$ and $a$ be a tangent ...
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2answers
194 views

Is a closed compact 2-Manifold that is embedded in euclidean 3-space always orientable?

I am sorry if this is a trivial question but I am a little confused right now so please bear with me. Since non-orientable compact 2-manifolds without boundary cannot be embedded in three-dimensional ...
2
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2answers
98 views

What does “flat hypersurface” mean?

If $S$ is a flat hypersurface with boundary in $\mathbb{R}^n$, what does it mean? Is it just a simple open domain (found in most PDE contexts)?