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

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A form of chain rule to differentiate the flow of a vector field on a manifold

I am reading the proof for a theorem about connections on a manifold, but I'm not comfortable with the fancy language of vector bundles and flows of vector fields I think. I wonder if there's an easy ...
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Gauss Curvature Equation

One can find the following form of Gauss Curvature Equation in most introductory books on manifold, for example Jeffrey Lee's Introduction to Differential Geometry p564: $$\langle R(V,W)X,Y\rangle ...
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Is $(X_G, d_G)$ a compact manifold?

Let $G$ a compact topological group act on $(X,d)$ by isometries. We define a relation $\sim$ on $(X, d)$ as follows: for $x,y\in X$: $$x\sim y \Leftrightarrow x=gy \ \text{ for some } g\in G.$$ It ...
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Let $M_1 , M_2$ two manifolds of dimension $n_1, n_2$ and $M_1 \subset M_2$. Prove: $M_1$ is relatively open in $M_2$ $\iff n_1 = n_2 $.

Let $M_1 , M_2$ two manifolds of dimension $n_1, n_2$ and $M_1 \subset M_2$. Prove: $M_1$ is relatively open in $M_2$ $\iff n_1 = n_2 $. I have no idea where to start with this one, any help would be ...
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A compact $n$-manifold is orientable iff there is an everywhere nonzero $n$-form

Let $M$ be a compact differentiable manifold of dimension $n$ without boundary. Show that $M$ is orientable if and only if there exists a diffential $n$-form $\omega$ defined on $M$ and which is ...
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Is a maximal open simply connected subset $U$ of a manifold $M$, necessarily dense?

There is a short argument using Zorn's lemma and the compactness of $[0,1]$, that shows every manifold must have maximal open simply connected subspaces. However, I am wondering if it is necessarily ...
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Tangent Bundle of the open positive orthant

What is the expression for the tangent bundle of the open positive orthant $\mathbb R_+^n$? I think I know the answer, but just to be sure. Thanks
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49 views

Actions of a finite group.

I've been playing around with this proof for a while and I can't seem to figure out where to go from here: I have that $M$ is a manifold, $G$ is a finite group (say, of order $n$), and the action of ...
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1answer
23 views

Show that $\mu_{∗,(e,e)}(X, Y ) = X + Y$ [on hold]

This is an assignment: (a)Show that $\mu_{∗,(e,e)}(X, Y ) = X + Y$ for any $ (X, Y ) ∈ T_eG ⊕ T_eG$ where $G$ is a Lie group? (b)Show that $ι_{∗,e}(X) = −X$ for any $X ∈ T_eG$ Can anyone help me ...
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25 views

Int M is open and a manifold

If M is an n-dimensional manifold with boundary, then Int M is an open subset of M , which is itself an n-dimensional manifold without boundary. I am supposed to use these definitions: If M is an ...
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26 views

Abbreviating the definition of a tangent vector field?

Let $A \subset \mathbb{R}^{n}$ be open in $\mathbb{R}^{n}$ and let $F: A \to \mathbb{R}^{n} \times \mathbb{R}^{n}$ be continuous. Then $F$ is called a tangent vector field on $A$ if and only if $F(x) ...
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28 views

Preimage of a submanifold is a submanifold - Transversality

It is well known that if a smooth Map $f : M \to N$ between two smooth manifolds (finite dimensional) is transversal to a submanifold $L \subset N, L \pitchfork f$, than $f^{-1}(N)$ is a submanifold ...
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15 views

Positive definite quadratic form . Riemannian manifolds

Does anybody know how to solve it? I've done a lot of tries but I didn't succeeded Let $H^n=\{ (x_0,x_1,...,x_n)\in \Re^{n+1}:x_0^2+x_1^2+...+x_n^2=-1,x_0>0\}$ and the symetric bilinear form ...
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15 views

Foliation vs Coordinates in de Sitter

I'm studying de Sitter manifolds and am confused about the difference between the choice of foliation and the choice of coordinates (and how they relate to the spatial curvature). I can choose the ...
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1answer
22 views

Dimension of unordered configuration space

I'm working on $X = \mathbb{R}^n$ and I'm considering the set of unordered sequence of points of $X$. Considering $F(p) = \lbrace (x_1, \dotsc, x_p) \in X^p ; i \neq j \implies x_i \neq x_j \rbrace$ ...
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Showing components of manifolds are open [on hold]

Show that the connected components of a manifold M are open sets and are countable in numbers.
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Definition of a length metric.

Let $(X, d)$ be a metric space. Define the induced intrinsic metric $\widehat{d}$ as follows \begin{equation} \widehat{d}(x,y) = \inf_{\gamma} \sup_{t_0, \ldots, t_N} \sum_{i=1}^{N}d(\gamma(t_{i-1}, ...
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Reference request: Tubular neighborhood theorem for non-closed manifolds via the exponential map.

Let $M\subset N$ be a submanifold. (Both $M$ and $N$ have no boundary, but $M\subset N$ need not to be closed as a subspace and none of them need to be compact) Choose a Riemannian metric on $N$ and ...
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37 views

Embedding of a topological manifold

We know the celebrated 'Whitney embedding theorem' for smooth manifold that says any n-dimensional manifold can be smoothly embedded in $\ \mathbb R^{2n} \ $. Now my question is: Is there similar ...
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1answer
43 views

How to calculate the tangent space?

first of all, what is the difference between the tangent space and the tangent plane? I tried to find the tangent space of the hyberpoloid $$x^2 +y^2 -z^2 =a$$ , $$a>0 $$ at the point ...
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Stable/Unstable Manifold heorem

Why does the stable/unstable manifold theorem imply that the power series expansion of the stable/unstable manifold is locally convergent? (local to the fixed point)
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Is the connected sum of complex manifolds also complex?

Let $M$ and $N$ be real manifolds of dimension $n$ which happen to admit complex structures (so that necessarily $n=2k$ and both are orientable). Then does their connected sum $M\# N$ also admit a ...
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1answer
48 views

2.25 of Lee's introduction to topological manifolds

If M is an n-dimensional manifold with boundary, then IntM is an open subset of M , which is itself an n-dimensional manifold without boundary. Here are the definitions to use: If M is an n-manifold ...
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2answers
67 views

Definition of Cech-De Rham complex. Can't understand its definition!!!!

Let $M$ be a manifold and let $\mathcal{U}:=\{U_\alpha\}_{\alpha\in I}$ be an open covering, $I$ be a totally ordered set. For every $p$ and for every $\alpha_0<\dots<\alpha_p$, $$ ...
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1answer
20 views

Transition matrix for coordinate

Suppose that $ (U,x^1,x^2,...,x^n) $ and $ (V,y^1,y^2,...,y^n) $ are two coordinate charts on a manifold.Then $$ {\partial \over \partial x^j}=\sum_i {\partial y^i \over \partial x^j } {\partial \over ...
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1answer
67 views

Manifold that is not a Euclidean space

I just started reading a textbook, and it keeps saying that an $n$-dimensional manifold is a topological space with the same local properties as Euclidean $n$-space. I don't really understand what is ...
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40 views

Complex-valued differential forms.

Let $X$ be smooth (real) manifold and let $T^{*}(X)_{\mathbb{C}}$ denote the complexification of the cotangent bundle. We define the complex valued differential r-forms on $X$ to be the smooth ...
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36 views

The second fundamental form of the sphere

I am trying to understand how one computes the second fundamental form of the sphere. Looking at the example on page 10. http://www.math.miami.edu/~galloway/dgnotes/chpt5.pdf Here I understood ...
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Exponential map on a sphere in spherical coordinates

Let $M = \{ (x^\varphi, x^\theta) : x^\varphi \in [0, \pi), \thinspace x^\theta \in [0, 2\pi) \}$ be a manifold with metric $\mathrm{d}s^2 = (\sin x^\varphi)^2 (\mathrm{d} {x^\theta})^2 + (\mathrm{d} ...
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Orientability and volume form

I was wondering if there is an easy argument that manifolds that are orientable always have a volume form and vice versa? So I am not looking for a full proof of this, but rather a good argument how ...
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135 views

What hyperbolic space *really* looks like

There are several models of hyperbolic space that are embedded in Euclidean space. For example, the following image depicts the Beltrami-Klein model of a hyperbolic plane: where geodesics are ...
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1answer
51 views

Specifying an arbitrary point on a manifold

It is known that any arbitrary point x on the sphere $\mathbb{S}^2$ can be parametrised by the spherical coordinates $$\bf{x}=r(\theta,\phi)=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta),\quad ...
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Advanced Differential Geometry Textbook

In algebraic topology there are two canonical "advanced" textbooks that go quite far beyond the usual graduate courses. They are Switzer Algebraic Topology: Homology and Homotopy and Whitehead ...
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183 views

Is paralellizability a topological invariant (Invariant under homemorphism)

This MO post is a motivation to ask: Is paralellizability a topological invariant (Invariant under homeomorphism)?
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217 views

How can one compare these two 4-manifolds

We would like to compare the following two real 4 dimensional manifolds: 1)$M$=The tangent bundle of $S^{2}$ 2)$N$= The total space of the canonical line bundle over $\mathbb{C}P^{1}\simeq S^{2}$ ...
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an isomorphism between the tangent space of a manifold to euclidean space

1) I was told in class many years ago that, the tangent space if the sphere $\mathbb{S}^2$at a point $p$, i.e. $T_p\mathbb{S}^2$ is isomorphic to $\mathbb{R}^2$. Could anyone give me a proof of ...
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53 views

Why this manifold can not be embbeded in 3d Euclidean space?

Consider the following system of 6 equations in 9 variables $x_1, x_2,x_3,...x_9$ $ x_1 ^ 2 + x_2 ^ 2 + x_3 ^ 2 + = 1 $ $ x_4 ^ 2 + x_5 ^ 2 + x_6 ^ 2 + = 1 $ $ x_7 ^ 2 + x_8 ^ 2 + x_9 ^ 2 + = 1 $ ...
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1answer
88 views

Can a fractal be a manifold?

Here it is said that it is not possible: Can a fractal be a manifold? if so: will its boundary (if exists) be strictly one dimension lower? But I am confused about this. What about the invariant ...
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85 views

Equivalent condition for non-orientability of a manifold

I've just came across this question, which gives us a great tool for showing that smooth manifold is non-orientable. Namely Thm. If $M$ is a smooth manifold and there are two charts ...
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1answer
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Can a single point in a manifold be seen as a sub manifold?

In Pollack's differential topology, in Transversality, p.28, it reduced the study of the submanifold $Z$ to the simpler case, where $Z$ is a single point. But by the definition of manifold, it seems ...
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1answer
72 views

flat manifold, curvature and the circle

A Riemannian manifold is said to be flat if the curvature is 0 everywhere. An example in dimension 1 is the circle. However, I cannot see how the curvature of the circle could be 0. See for instance ...
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product of spaces is a manifold. Are the spaces?

Suppose that $X$ and $Y$ are topological spaces and that $X\times Y$ is a topological manifold. It seems that we can't conclude that $X$ or $Y$ are manifolds themselves (this question). EDIT :Are ...
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Simpler version of dogbone space construction

In "The cartesian product of a certain nonmanifold and a line is $E^4$" (R.H. Bing, Annals of Mathematics series 2 vol 70 1959 pp. 399–412) Bing constructs a nonmanifold, $B$, such that $B\times \Bbb ...
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Definition of complex submanifold

For smooth manifolds, we can define an embedded submanifold to be either (1) a subset locally cut out by "slice" charts, or (2) a subset that is a manifold in the subspace topology and admits a smooth ...
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1answer
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Derivatives and the cotangent space

In Differentiable Manifolds, the derivative of a function $f: M \rightarrow \mathbb{R}$ at $a$ denoted by $(df)_a$ is defined as its image in the cotangent space: $T_a^* = C^\infty(M)/Z_a$, where ...
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Reference request: Pullbacks along submersions are submanifolds and the induced map is a submersion

I'm looking for an introductory book in differential topology in which there are proofs of the following facts: Let $X,X',Y$ be smooth manifolds, $X\rightarrow Y$ a smooth map and $X'\rightarrow Y$ ...
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Is the restriction of a maximal atlas on an open submanifold maximal?

Let $M$ be a $n$-manifold, with some maximal atlas $A$, and let $V \subset M$ be an open set. The standard open-submanifold-atlas on $V$ is $A|_V$ defined as $$A|_V = \big\{ (U \cap V,x|_{U \cap V}) ...
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1answer
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Restriction of differential $1$-forms to open subsets?

A vector field on a manifold $M$ is a linear map $X:C^\infty(M)\longrightarrow C^\infty(M)$ with an additional property. The set $\mathfrak{X}(M)$ of all vector fields on $M$ is a ...
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Is the sphere $S^2$ diffeomorphic to a quotient of the square?

If we take the square $[0,1]\times [0,1]$ and collapse the border, the resulting quotient space is homeomorphic to the sphere. The same holds if we take the square $[0,1]\times [0,1]$ with the ...
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The definition of a functional structure on a topological space

I am reading Glen E. Bredon's Topology and Geometry and trying to solve the following problem in the book. But I can't understand what the blue g means. Could anyone explain please? Show that a ...