Differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.

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Frobenius theorem

I came across the following conclusion in a textbook, but can't really understand it. I would be grateful if anyone could elaborate: Assume that we have two linearly independent vector fields ...
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Vector field on n-manifold whose sum of indexes is equal to Euler charasteristic

For 2-manifolds and 3-manifolds such a tangent field (whose singular points indexes sum to manifold's Euler chracteristic) construction can be done visually. For example, for triangulated 2-manifold ...
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open set in tangentspace induces open set in tangentbundle (for homogeneous spaces)

Let $M=G/K$ be a homogeneous space with a $G$-invariant riemannian metric $<.,.>$. Then $G$ defines an action on $TM$ by derivatives. Let $p=eK \in M$. Assuming I have a set $V_p \subset T_pM$ ...
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Pushforward of a vector

The push forward of vectors allows to transform the components of a vector $X$ to be pushed along a map $h:\mathcal{M}\to\mathcal{N}$ between the manifolds $\mathcal{M}$ and $\mathcal{N}$. This is ...
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Condition for set of de Rham cohomology classes is linearly independent

If we have a set of 1-forms $w_1, ... w_n$ on a smooth manifold $X$ I can show that $w_i$ are linearly dependent if and only if $w_1 \wedge ... \wedge w_n = 0$. I wondered if this is also true in the ...
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How can we define $\partial x_{i_r}^p(X_p^r)$?

Let $M$ be a smooth manifold and $X_r^s:M\to T_r^s(M)$ is a section. Let $P\subseteq M$ be an open set and $X_1,\ldots,X_s\in\mathcal T(P)$ and $X^1,\ldots,X^r\in\mathcal T^*(P)$ where $\mathcal T(P)$ ...
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Explicit Morse functions on closed surfaces

Let $M_g$ a surface of genus $g$, why can we find a explicit Morse function on $M_g$? For the torus we have $f(x,y)=\cos(2\pi x) + \cos(2\pi y)$, for sphere $f(x,y)=x^2+2y^2$ and this induces an ...
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A lemma in Milnor's book “Topology from the Differentiable Viewpoint”

In preparation for proving the Poincaré-Hopf Theorem, Milnor states in his book (see p. 38, Theorem 1) For any vector field $v$ on $M$ with only nondegenerate zeros, the index sum $\sum \iota$ ...
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Understanding algebraic operations on vector bundles

Let $X$ be a smooth manifold with vector bundles $V$ and $W$. I'm trying to understand how we can construct in general a vector bundle $V \otimes W$. In particular what manifold structure do we give ...
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Compact-Open Topology for Space of C^{r} -sections

Given a smooth fibre bundle $\pi: X \rightarrow M$. What is the definition of compact open $C^{r}$-topology on the space of $\mathcal{C}^{r}$-sections?
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Cohomology of local systems on spaces with the same homotopy type

Suppose we have a homotopy equivalence $f: X \to Y$ with homotopy inverse $g: Y \to X$, and a local system (i.e. a locally constant sheaf) $\mathcal{V}$ on $Y$. In this situation, are any of the ...
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Low torsion in orientable manifolds?

The final sentence on page 170 of Stillwell's Classical Topology an Combinatorial Group Theory is: Poincaré justified the term "torsion" by showing that $(m-1)$-dimensional torsion is present only ...
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Transversality: what is wrong with this counter example to persistence for small perturbations?

Let $M$ and $N$ be differentiable manifolds in $\mathbb{R}^{n}$, and let $p \in \mathbb{R}^{n}$. We say that $M$ and $N$ are transversal at $p$ if $$T_{p} M + T_{p}N = \mathbb{R}^{n}.$$ By dimension ...
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Parallel transport perspective of gauge transformation invariance for connections

Defining a connection on a principal $G$-bundle $P \to M$ is equivalent to defining a parallel transport on $P$ along curves in $M$. With this perspective, Ralph Cohen commented in his notes on the ...
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What is the tangent space o SO(n) [closed]

It should be the kernel of the map $H\mapsto A^TH+H^TA$ at some $A$ such that $A^TA=I$ . But I cant find this Kernel can someone help me?
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Fundamental group - space of copies of circle $S_1$

For $n>1$ an integer, let $W_n$ be the space formed by taking $n$ copies of the circle $S_1$ and identifying all the $n$ base points to form a new base point, called $w_0$. What is $\pi_{1}$($W_n , ...
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Immersions-possible error in Dieudonné III?

Below I refer to [D] Dieudonné Treatise on analysis III [B] Bourbaki VARIETES DIFFÉRENTIELLES ET ANALYTIQUES [M] Michor Topics in differential geometry In [D,16.7.7], we can read: Let $f ...
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Do compact connected smooth manifolds admit the structure of a CW complex with a single 1-cell? [closed]

This seems intuitive to me, since they admit a CW decomposition with finitely many cells. But I can't see how to prove it.
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On Atiyah-Singer Index theorem and Atiyah-Bott fixed point theorem.

I would like to understand the statement of Atiyah-Singer and Atiyah-Bott theorems enough to see lots and lots of applications they have. My background is pretty thin with some basic algebraic ...
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Reference request: Inclusion of smooth maps into continuous maps between smooth manifolds is a weak homotopy equivalence.

Let $M,N$ be smooth manifolds. It seems to be well known that if the sets $C^0(M,N)$ and $C^\infty(M,N)$ are equipped with the appropriate topologies (I suppose the weak/strong Whitney topology), then ...
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$SO(n)$ is connected

The question really is that simple: Prove that the manifold $SO(n) \subset GL(n, \mathbb{R})$ is connected. it is very easy to see that the elements of $SO(n)$ are in one-to-one correspondence with ...
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Orientation on the boundary

If $M$ is an oriented without boundary manifold, and $\mu$ is it volume form, is true that the boundary of $M\times [0,1]$ is $ M \cup M$, right? It is true also that the orientantion on the boundary ...
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Proving Euler's spiral is an isometric embedding with bounded image

$\newcommand{\al}{\alpha}$ I am trying to prove Euler's spiral is an isometric embedding of $\mathbb{R}$ into $\mathbb{R}^2$ with bounded image. Here is the definition of the spiral: $(*) \, ...
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Degree of smooth map of manifolds depends on orientation choice?

I'm a little to confused as to why it appears that the degree of a smooth map $f: M \to N$ between smooth manifolds appears to only be defined up to sign - I'm not sure where my mistake is. By ...
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Klein bottle in $\mathbb{R}^4$ does not have a couple of normal vector fields

For an orientable 2-manifold in $\mathbb{R}^4$, it's somewhat obvious that if the manifold has a normal vector field then it has a pair of normal vector fields. I am trying to understand why it is ...
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Minimal requirements to be a submersion.

I saw here (A surjective map which is not a submersion) that a smooth differentiable map $f:M\to N$ between two manifolds $M$ and $N$ is not necessarily a submersion. A counterexample is ...
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degree 1 map $f : M \to S^n$

Let us consider $M$ a closed and connected n-manifold which is also orientable. An exercise in Hatcher claims that for any such $M$ there is a continuous map $f: M \to S^n$ such that it's degree is 1, ...
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Fundamental class for tangent bundle

For closed orientable manifolds $X$ we can define the fundamental class $[X]$ which is just a choice of generator of the top homology group $H_n(X; \mathbb{Z})$. However, in the context of the ...
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Is every self-homeomorphism homotopic to a diffeomorphism?

Given a smooth manifold $M$, is every homeomorphism $M \to M$ homotopic to a diffeomorphism? Hirsch's "Differential Topology" has a proof that every $C^1$ diffeomorphism of smooth manifolds is ...
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Why do people care about principal bundles?

I've started to learn a little about principal bundles (in the smooth category) and while I see how notions like connections and curvature are abstracted from the setting of vector bundles and brought ...
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Isomorphic modules of sections imply isomorphic bundles

For reference this is about a part of question 3-F in Characteristic Classes by Milnor and Stasheff, which discusses the $C(B)$-module structure of the space of sections of a topological vector bundle ...
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Every real matrix with non-negative entries has a non negative eigenvalue [duplicate]

If $A$ is any matrix $n\times n$ with non negative entries, then $A$ has a non negative eigenvalue. I know that I have to use the Brower Point fix theorem, but I am not finding the function for that. ...
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Can every Riemmanian Manifold be completed?

I had two trails of though.. is either of them fruitful? I know every metric space can be completed, my question is: can a Riemmanian manifold $M$ be embedded smoothly and isometrically into it's ...
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Endomorphism vector bundle isomorphic to the adjoint bundle of its frame bundle?

Could somebody help me to prove the following isomorphism (in particular what is the isomorphism)? \begin{equation} End(\xi) \cong ad(E_{\xi}) = E_{\xi} \times_{GL(n,\mathbb{R})} ...
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partial differentiation on a manifold

im currently taking up a course on differential geometry and the last topic is topology. id like to ask for help in our homework since im kind of new to this kind of questions which involves proving ...
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$\operatorname{SU}(n)$ as manifold

I am trying to do this has a while, but I cannot use correctly the regular value theorem to do so! I appreciate any help. The problem is that I cannot choose the function to take $SU_n$ as a regular ...
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Mobius strip as manifold and as a bundle over $S^1$

I am trying to construct the Mobius strip bundle onver $S^1$. I am stuck in how to define charts for the Mobius bundle. Once I make this, the problem will easily be solved. My attempt was: $$M = ...
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If $f: A \subset \mathbb{R}^m \to \mathbb{R}^n$ is of class $C^1$ and in $a \in A$ rank of $f$ is $p$ there is an embedding

If $f: A \subset \mathbb{R}^m \to \mathbb{R}^n$ is of class $C^1$ and in $a \in A$ rank of $f$ is $p$ there is an embedding $\phi : V \to A$, of class $C^{\infty}$ such that $f\circ \phi$ is an ...
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How much of an $n$-dimensional manifold can we embed into $\mathbb{R}^n$?

I observed some naive examples. Spheres, for example, when we cut out one point, can be embedded into $\mathbb{R}^n$. And if we cut out a measure zero set of a projective space, it can be embedded ...
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Local gauge transformation law on a principal bundle

I am referring to the answer by Henry to a related old question. Since it has been a long time I post it up as a new question instead of appending to the old one as a comment. The local gauge ...
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1answer
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Why is $x^{1/3}$ not differentiable?

The problem says On $\mathbb{R}^1$consider $f(x)=x$ and $g(x)=x^{1/3}$ both $\mathbb{R} \to \mathbb{R}$. Consider atlases $\alpha_1=\{(\mathbb{R},f)\}$ and $\alpha_2=\{(\mathbb{R},g)\}$. Show that ...
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Show that piecewise function $f$ is $C^{\infty}$

I don't understand the first line of the solution If $f:\mathbb{R} \to \mathbb{R}$ defined by $f(x)={e^{-1/x}}$ if $x>0$ and $f(x)=0$ if $x \leq 0$ then show it is $C^{\infty}$. Well, it frst ...
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Showing deformation retract : $B^3/T^2$, $R^3/S^1$, $S^2 \wedge S^1$

Here what i want to show is $B^3/T^2$, $R^3/S^1$, $S^2 \wedge S^1$, $i.e$, three spaces are deformation retract to each other. Can you give me some hints or concept(?) geometric way to show this? ...
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Number of non-homotopic diffemorphism form a manifold to itself

What is the name of this invariant, the number of non-homotopic diffemorphism form a manifold to itself. What is this number for the closed ball B^n, and for euclidean space R^n and for the n-sphere?
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Euler characteristic of branch cover of punctured Riemann surface

Let $\Sigma_1$, $\Sigma_2$ be two closed Riemann surfaces, $\pi: \Sigma_1 \to \Sigma_2$ is degree $m$ branched cover of $\Sigma_2$, then we have formula about their Euler number: $$\chi(\Sigma_1)= ...
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What are the essential tools and proof techniques for beginning smooth manifolds and differential topology?

I am an undergraduate currently taking a first course in smooth manifolds. I feel that I understand the material intuitively. But, I'm having trouble turning my intuition into proofs. I was hoping ...
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Given $p \in S^{n-1}$, how does one show that the map $\pi: SO_n \to S^{n-1}$ by $A \to A(p)$ is a submersion?

Pick $ p \in S^{n-1} \subset \mathbb{R}^n$ and consider the map $\pi: SO_n \to S^{n-1}$ by $A \to A(p)$. Show that this map is a smooth submersion. For $ q \in S^{n-1}$, describe the pre-image. For ...
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Index of a Jordan curve

Winding number theorem: If $J\subset \mathbb{C}$ is a Jordan curve and a point $z$ lies in its interior domain, then the winding number $n(J,z)=\pm 1$. Now suppose that $J$ is smooth and we have the ...