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

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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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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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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
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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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66 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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1answer
39 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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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 $\mathrm{d}s^2 = (\sin \varphi)^2 \mathrm{d}\theta^2 + \mathrm{d}\varphi^2$ ...
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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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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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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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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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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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1answer
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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82 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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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
35 views

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
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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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1answer
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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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1answer
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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 ...
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Integration of densities?

I am stuck with a problem about integration on densities. Given is a density $\sigma (X,Y) =\frac{2}{(X^{2}+Y^{2}+1)^{2}}$ on $S^{2}$ sphere. Compute $\int_{S^{2}}\sigma $. The answer in my book is $2 ...
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Is this a correct way of thinking about diffeomorphic manifolds?

In set theory there is the concept of a bijection, a one-to-one correspondence between the elements of $2$ sets. In topology the concept of a homeomorphism $f:X\to Y$ is quite easy to wrap your head ...
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Do two embeddings of a Euclidean space into a higher dimensional one only differ by a diffeomorphism?

Let $d\le n$ and $$f,g\colon\mathbb{R}^d\hookrightarrow\mathbb{R}^n$$ be two smooth embeddings. Is there a diffeomorphism $$\phi\colon\mathbb{R}^n\rightarrow \mathbb{R}^n,$$ such that $$f=\phi\circ ...
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Formal adjoint of curvature (Yang Mills)

Currently reading a paper on finding solutions to the Yang Mills equation $D^*\Omega=0$, where $\Omega$ is the curvature and $D^*$ is the formal adjoint of the exterior covariant derivative $D$. ...
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1answer
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Proof that a function $f:\mathbb{R}\times\mathbb{R}\to N$ restricts to a smooth function on $S^1$

I have to prove Proof that a function $f:\mathbb{R}^2\to N$ (where $N$ is a smooth manifold) restricts to a smooth function on $S^1$ Here $S^1$ is defined as the subset of $\mathbb{R}^2$ ...
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A detail in the proof orientable manifold admits exactly two orientations

Let $M$ be a real manifold and let $\{(U_\alpha,\phi_{\alpha})\}_{\alpha\in I}$ $\{(V_\beta,\psi_\beta)\}_{\beta\in J}$ be two oriented atlases. Let's define, for $p\in U_\alpha\cap V_\beta$, ...
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1answer
32 views

“Winding number”, Chern character and relative signatures of the metric

Anyone answer with good explanation is appreciated. In differential geometry, we discuss about topological quantities like characteristic classes. For example, the first Chern character of some ...
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Is the signature of inverse images of diffeomorphic submanifolds (along a homotopy equivalence) the same?

Suppose it is given an orientation preserving homotopy equivalence $h:N\to M$ between closed oriented connected manifolds. Let $X$ and $Y$ be diffeomorphic submanifolds of $M$, and assume $h$ to be ...
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1answer
35 views

Derivation of tensor components transformation in tangent space

Might anyone offer a derrivation? My attempt bellow ($ x_{i'}$ is counting through the transformed coordinates) $\displaystyle\frac{\partial }{\partial x_{i'}}= \displaystyle\frac{\partial ...
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Closed orientable three manifold with finite cover by $S^1 \times S^2$ or $T^3$

I have been thinking about a problem where I can conclude that I have a closed orientable three manifold which is covered by $S^1 \times S^2$ or $S^1 \times T^2$. I think that the geometrization ...
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1answer
34 views

A function $f: \Bbb R \to \Bbb R$ which is differentiable at a point but not $C^1$ at that point.

Give me an example of a function $f: \Bbb R \to \Bbb R$ which is differentiable at a point but not $C^1$ at that point. I have found a function $f: \Bbb R^2 \to \Bbb R$, $f(x,y)=|xy|$ which is ...
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Open balls in the definition of a Euclidean submanifold

Stroock (Essentials of Integration Theory For Analysis, $\S8.3.4$) defines a $k$-dimensional submanifold of $\mathbb{R}^n$ to be a set $M \subseteq \mathbb{R}^n$ such that for all $x \in M$, there ...
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1answer
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“Error”/“Gap” in proof on every topological manifold being regular

I define topological manifold as in Munkres: Hausdorff, second countable, and every point has a neighborhood homeomorphic to an open subset of Euclidean space. My definition of regular is as in ...
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1answer
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$X$ is connected and $\mathcal{U}$ is a locally finite collection of precompact open sets that covers $X$; then $\mathcal{U}$ has a countable subcover

I came to this question in the course of trying to prove that if $X$ is a paracompact locally Euclidean Hausdorff space with at most countably many components, then it is second countable. I was able ...
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1answer
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$N$ commuting vector fields on an $N$-dimensional compact manifold

If an $N$-dimensional compact manifold has $N$ commuting vector fields, does this mean the manifold is actually a torus?
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1answer
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A compact, connected, abelian Lie group is a torus?

How to prove that a compact, connected, abelian Lie group is a torus? It seems very intuitive. Any reference?
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1answer
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Does the fixed point in the Brouwer's Fixed-Point Theorem have to be an interior point?

As the question suggests, does the fixed point in the Brouwer's Fixed-Point Theorem have to be an interior point? Many thanks in advance. To be clear, I'm using the statement of Brouwer's Fixed-Point ...
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Manifold, exist smooth nonnegative function with regular value at 0?

If $X$ is any manifold with boundary, then does there exist a smooth nonnegative function $f$ on $X$, with a regular value at $0$, such that $\partial X = f^{-1}(0)$?
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Intersection of topological manifolds.

A condition for the intersection of two smooth manifolds to be a smooth manifold is that they intersect transversally. Is this only an obstruction because of the smooth structure? Question: Is the ...
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Compute the tangent space at the unit matrix

Compute the tangent space $T_pM$ of the unit matrix $p=I$ when $$(i)\,M=SO(n)\\ (ii)\,M=GL(n)\\ (iii)\,M=SL(n).$$ My attempt: I think I have computed the tangent space in the case that $M=SL(n)$. ...