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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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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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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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) [on hold]

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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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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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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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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Do compact connected smooth manifolds admit the structure of a CW complex with a single 1-cell? [on hold]

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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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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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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Is there any difference between Immersion and embedding?

Definition as below , I think they are same ,is right ?
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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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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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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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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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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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How can we define $\partial x_{i_r}^p(X_p^r)$?

Suppose $M$ is a manifold and $X_r^s:M\to T_r^s(M)$ is a section. Let $P$ 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 $\mathcal T(P)$ and $\mathcal ...
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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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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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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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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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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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Such a manifold is homeomorphic to a sphere

I recently read that if a compact differentiable manifold admits a real function with only two critical points, then it is homeomorphic to a sphere. If the function is Morse, this follows from ...
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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 ...
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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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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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A Riemannian metric on the torus $T^n$

This exercise is from Do Carmo, Riemannian Geometry. Introduce a Riemannian metric on the torus $T^n$ in such a way that the natural projection $\pi:\mathbb{R}^n\to T^n$ given by ...
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Bott&Tu Definition: "Types of Forms:

In Bott&Tu's well-known book "Differential forms in Algebraic topology", they note -(p34): every form on $\mathbb{R}^n \times \mathbb{R}$ can be decomposed uniquely as a linear combination of two ...
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Tangent bundle of sphere as a complex manifold

I'm trying to show that the tangent bundle, $TS^n$ of the n-sphere $S^n$ is diffeomorphic to the set $\sum z_i^2 = 1$ in $\mathbb{C}^{n+1}$. It's relatively straightforward to see that the tangent ...
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Does every $C^1$ closed differential form differ from some $C^\infty$ closed form by an exact form?

Question: On a $C^\infty$ manifold, does every $C^1$ closed differential form differ from some $C^\infty$ closed form by an exact form? Motivation: This result holds for $C^1$ closed 1-forms on a ...
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Connectedness of the level sets of Mose-Bott functions

I am reading the proof by McDuff and Salamon of the connectedness of the level sets of Morse-Bott functions with index and coindex different from $1$: connectedness of level sets A smooth function ...
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Is a translation in a compact Lie group homotopic to the identity?

The following exercise is from Guillemin and Pollack, Differential Topology. Show that the Euler characteristic of the orthogonal group (or any compact Lie group for that matter) is zero. Hint: ...
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Help finding smooth functions that agree on the boundary, but avoid a critical value.

Basically let $U$ be something like a compact neighborhood of $\mathbb{R}^n$ with smooth boundary $\partial U$ and suppose $f:U \rightarrow \mathbb{R}^n$ is smooth. Now fix $x_0 \in \mathbb{R}^n$ with ...
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The quotient of a manifold by a submanifold is never a manifold?

Let $M$ be a connected smooth manifold. Let $S$ be a connected embedded submanifold of positive dimension and co-dimension, which is also a closed subset of $M$. Is it true that the quotient space ...
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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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Can a connected finite-dimensional manifold have cardinality $>2^{\aleph_0}$?

Can a connected finite-dimensional manifold have cardinality $>2^{\aleph_0}$? I know that if we either impose the condition "Hausdorff" or "second countable", the assertion is false. What if ...
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Can a connected second countable manifold have cardinality $>2^{\aleph_0}$?

Can a connected second countable manifold have cardinality $>2^{\aleph_0}$? From here, a connected Hausdorff manifold must have cardinality $2^{\aleph_0}$. How about if we change the condition ...
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Every compact hypersurface in $\mathbb{R}^n$ is orientable

Show that every compact hypersurface in $\mathbb{R}^n$ is orientable. HINT: Jordan-Brouwer Separation Theorem. This is an exercise from Guillemin and Pollack. So hypersurface means smooth ...
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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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Exterior derivative as a special case of covariant derivative?

In terms of local coordinates we can make a covariant derivative exterior-derivative-like (this is actually Levi-Civita connection) \begin{equation} \Gamma_{i,j}^k = \Gamma_{j,i}^k \implies D(dx_k) ...