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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Subdifferential of a convex function

How would I find a convex function $f: \mathbb{R} \to \mathbb{R}$ such that $\partial f(0) = [0,1]$ A subdifferential is just the collection of vectors $w \in \mathbb{R}^n$ such that $f(y) \geq ...
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Let C be an oriented plane curve with curvature k>0. Assume that C has at least one point p of self intersection, prove the following questions:

Let C be an oriented plane curve with curvature k>0. Assume that C has at least one point p of self intersection, prove the following questions: a. There is another point $p_{0}$ such that the ...
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Immersion, embedding and category theory

The map between smooth manifolds $f:M \to N$ is called an immersion if the tangent map $Tf$ is injective for each $x \in M$. There is a corresponding notion of submersion. About embeddings I met two ...
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Immersion $\mathbb{S}^n\times\mathbb{R}\to\mathbb{R}^{n+1}$

Immersion $\mathbb{S}^2\times\mathbb{R}\to\mathbb{R}^3$ As $\mathbb{S}^2\times\mathbb{R}$ it's not compact, i can give immersion given by $$(x,y,z,t)\to e^t(x, y, z).$$ or i'm wrong. Could you give ...
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Interpretation of local submersion theorem

Wikipedia gives the following formulation of the $\textbf{Local Submersion thoerem}$, If $f: M \to N$ is a submersion at $p$ and $f(p)=q \in N$, then there exists an open neighborhood $U$ of $p$ in ...
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Embedding the mapping $\phi:\mathbb{CP}^n\times{\mathbb{CP}^m} \to\mathbb{CP}^{nm+n+m}$

The canonical coordinates of $\mathbb{CP}^n$ are $x=[(x_0,\ldots, x_n)]$, $x_i\in\mathbb{C}$. How to prove that the mapping $$\phi:\mathbb{CP}^n\times{\mathbb{CP}^m} \to\mathbb{CP}^{nm+n+m}$$ ...
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Manifold diffeomorphic to $\mathbb{S}^1\times\mathbb{R}$.

Let $H=\{(x, y, z)\in\mathbb{R}^3 | \dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}−\dfrac{z^2}{c^2}=1\}$. Prove that $H$ is a manifold diffeomorphic to $\mathbb{S}^1\times\mathbb{R}$. Already I tried to be spread ...
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ways to see whether the Pontryagin class of a quaternionic line bundle over a CW-complex is zero

the first pontryagin class of a quaternionic line bundle over a CW-complex is zero if and only if the quaternionic line bundle is trivial or not? Let $\xi^\mathbb{H}$ be a given quaternionic line ...
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Question about Milnor's proof of Sard's Theorem

We've just covered Sard's theorem and have just started to look at transversality in my differential geometry class and I'm trying to understand a proof of Sard's theorem (based on Milnor's proof): If ...
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Open cover of manifold with boundary

If $\mathcal{O}=(U_{\alpha})_{\alpha\in A}$ is an open cover for a smooth manifold $M$, then each $U_{\alpha}$ is a smooth manifold. I want to extend this fact to manifolds without boundary. So my ...
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Winding number integral/index in plane

Let $B(x_0,y_0)$ be an open unit disk. Assume that $F(x,y)=(f_1 (x,y),f_2 (x,y)):\overline{B(x_0,y_0)}\rightarrow \mathbb{R}^2$ is a diffeomorphism. Assume also that $F(x,y) \ne (s_0,t_0) $, for all ...
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When did the notion of the tangent space emerge?

When did the modern conception of/notation for the tangent space to a manifold come into use?
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35 views

Mapping degree of a diffeomorphism

This might be a bit silly question but I haven't find direct reference. Let $\Omega$ be open, bounded and connected in $\mathbb{R}^n$. Assume that $f:\overline{\Omega}\rightarrow \mathbb{R}^n$ is a ...
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Smooth manifolds having an analytic structure.

I am reading about Manifolds and Lie groups and I have certain questions related to them. Let me first explain why I am asking these questions. We know that not all $C^{\infty}$ functions are ...
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chern class of complex line bundle

Let $\xi:=(\mathbb{C},E,p,B)$ be a complex line bundle, where $B$ is a manifold or CW-complex. How to determine whether the first Chern class $c_1(\xi)=0$ or non-vanishing?
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Lemma on locally finite open covers

I came across this lemma in Lee's 'Introduction to Smooth Manifolds'. The lemma seems simple enough to prove, but I just can't seem to prove it. It's frustrating me because I know it must be simple. ...
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integral cohomology of real grassmannians

I have obtained that the cohomology rings $$ H^*(G_k(\mathbb{R}^\infty);\mathbb{Z}_2)=\mathbb{Z}_2[w_1,\cdots,w_k]. $$ Also $$ H^*(G_k(\mathbb{R}^m);\mathbb{Z}_2)=\mathbb{Z}_2[w_1,\cdots,w_k]/(\bar ...
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Theorem 3.1 from Milnor's Morse Theory

Milnor is in the business of proving that if $f: M \to \mathbb{R}$ is a smooth function, $a < b$, and $f^{-1} ([a,b])$ is a compact subset of $M$ containing no critical points, then $M^a$ is ...
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63 views

Boundary on a manifold

I was wondering how I can see if a manifold has a boundary just by looking at the surface? The thing is that I want to understand how to apply the Gauß Bonnet theorem to surfaces and there I need to ...
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Derivatives in Topological Vector Spaces and General Spaces

I know that we may have a function $f$ such that all the directional derivative of $f$ in some topological vector space $V$, but the derivative need not exist. Moreover, it's possible for all the ...
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Is every $S^1$ knot orientable?

let $K$ be an $S^1$ knot, i.e., $K$ is a homeomorphic copy of the "standard" $S^1$ living in $\mathbb R^3$ ). I am trying to see if $K$ is orientable. This is what I have: 1) If the map sending $S^1$ ...
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90 views

Universe as a finite 3-manifold without boundary

My question is soft and imprecise, as I know very little differential topology. Nevertheless, I hope it makes some $\epsilon>0$ of sense. Assume the Universe is a 3-manifold without boundary, ...
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139 views

Invertibility theorem on the boundary for a function between two closed 2D manifolds

Assume a function $f:\mathbb{R}^2\to\mathbb{R}^2$ on a simply connected, closed domain $D\subset\mathbb{R}^2$ including its boundary $\partial D$. I am interested in the local invertibility of $f$ ...
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What is the anticommutator of the interior product and codifferential (adjoint of exterior derivative)?

What is $\eta=i_X \delta + \delta i_X$ acting on differential forms? Here $\delta$ is the usual Hodge adjoint of the exterior derivative and $i_X$ is contraction of a form with the vector field $X$. ...
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Map between Tangent Manifolds Well-Defined?

Let $f: \mathcal{M} \to \mathcal{N}$ be a $\mathscr{C}^{r+1}$ map. We define a map $\mathscr{T}f: \mathscr{T}\mathcal{M} \to \mathscr{T}\mathcal{N}$ as follows: A local representation of the map ...
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71 views

Why does the slope of a smooth simple closed curve have winding number one?

$\def\RR{\mathbb{R}}$Let $S^1$ be the circle and let $\gamma : S^1 \to \RR^2$ be a smooth injective map with $\gamma'(t)$ everywhere nonzero. What is the easiest way to show that $t \mapsto ...
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51 views

Ambient isotopy of based surface knots

Let $S$ be a smooth closed surface of genus $\ell$. Let $p$ be a point of $S$ and $a_i$, $b_i$ with $i=1,\ldots,\ell$ be $2\ell$ curves embedded in $S$ based at $p$ smooth everywhere except perhaps ...
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Stokes or homotopy?

The problem states Show that if $X$ is a simply connected manifold, then $\oint_{\gamma}\omega=0$ for all closed 1-forms $\omega$ on X and all closed curves $\gamma$ in $X$. However I have ...
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Embeddings are precisely proper injective immersions.

We call a map $f: X \to Y$ between topological spaces proper if $f^{-1}(K)$ is compact for all compact $K \subset Y$. Where can I find a reference that embeddings are precisely proper injective ...
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Embed curves in the plane

The strongest version of Whitney's embedding theorem says that every smooth real $n$-dimensional manifold $M^n$ (Hausdorff and second-countable) can be embedded in $\mathbb{R}^{2n}$. This should mean ...
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Transversality of leaves to the spheres .

Consider a form in the complex plane such that its linear part is $\omega_0=\lambda_1xdy-\lambda_2ydx$ in the Poincare domain: $\lambda_1\lambda_2 \ne 0$ and $\lambda_1/\lambda_2 \notin \mathbb{R}^-$. ...
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Laplacian and Hodge Laplacian

I am new to the theory of differential forms, but there is one thing that I don't get at all. Imagine that you are on the sphere $\mathbb{S}^2$, then the Laplacian $- \Delta$ is known to be a ...
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a generalization of punctured cylinder

Let $S^1\times \mathbb{R}$ be the infinite cylinder. Pucture it, we have $S^1\times \mathbb{R}-*$. Then $(S^1\times \mathbb{R}-*)\simeq Skeleton^1(S^1\times \mathbb{R})\simeq S^1\vee S^1$. How ...
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Terminology question : “half smooth, half topological” fibre bundle

First, I know (or I think I know...) the definition of fiber bundle, be it in the smooth or topological category. Here is my situation, which is kind of between the two: I have a smooth manifold $E$, ...
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cohomology ring of a quotient space

what is the cohomology ring $$ H^*((S^3\times S^3\setminus \{e\})/(a,b)\sim (ab,b^{-1});\mathbb{Z}_2)? $$ Here the unit $e$ and the product $ab$ is of the Lie group $S^3=Sp(1)$.
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Why are contact structures studied from a cohomological, rather than homological, perspective?

As far as I know, contact structures are studied from a "cohomology/dual" perspective, meaning mostly from the perspective of contact forms and their respective kernels, instead of from a ...
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Convention of a continued fraction presentation of a lens space

I want to clarify the following two conventions on a surgery description of a lens space. Let $p$ and $q$ are relatively prime integers. Express $$ ...
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Relation between differentials of perturbations of vector fields

Let $A$, $B$ be smooth submanifolds of a smooth manifold $M$ and $X\in C^\infty(TM)$ a vector field such following its flow $\xi^X$ gives a diffeomorphism between $A$ and $B$. Suppose also that ...
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Volume of a paracompact manifold

It is stated, without proof, in Wald (1984) (General Relativity) that given any connected manifold $M$ (which is by definition paracompact), one may define a volume measure $\mu$ such that $\mu[M]$ is ...
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Why any short exact sequence of vector spaces may be seen as a direct sum?

This is actually the first time I have worked with short exact sequences and I have no clue why the following assertion is true: Any short exact sequence of vector spaces $$ 0 \longrightarrow U ...
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Differentiability of a map into $\wedge T^*M$

Let $M$ a differentiable manifold and $p \in M$. Consider $(U;x_1, \dots, x_n)$ a coordinated system around $p$. $\{dx_\phi\}_\phi$ with $\phi \in \cal{P}(\{1, \dots n\})$, $dx_\phi=dx_{i1} \wedge ...
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Charts of $\wedge T^*M$ and $\wedge TM$.

Let $M$ a differentiable manifold and consider $\wedge T^*M$. If $(U, \phi)$ is a chart of $M$ with coordinate functions $(x_1, \dots, x_n)$, then $\{ \frac{\partial }{\partial x_i}\}_i$ and $\{ ...
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A question about a part of the proof of the existence of the exterior derivate.

Let $M$ a differentiable manifold, $(U;x_1,\ldots,x_n)$ a coordinated system and $\omega$ a differential form which domain intersects $U$. In $U \cap \operatorname{Dom} \omega$, $\omega$ can be ...
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25 views

Schwartz rule in differentiable manifolds.

Let $M$ a differentiable manifold and $(U,\varphi)$ a chart with coordinate functions $(x_1,...,x_n)$. Let $p \in U$. Given $f:U\longrightarrow \mathbb{R}$, $f \in C^\infty(U)$, it is possible to ...
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Explicit Calculation of the Euler class for the 2-Sphere using transition functions

I have been trying to learn about characteristic classes for months now, and every time I try a simple example something goes wrong. Any insights would be greatly appreciated. I am trying to follow ...
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59 views

Integral Curve of the vector field

If we have a 2-sphere with coordinates $x=r \cos \theta \cos \phi$, $y= r \sin \theta \sin \phi$ and $z=r \sin \theta$ and the vector field $X= (-r\sin \theta \cos\phi, r \cos \theta \sin \phi, r \cos ...
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Fundamental group of cusp of a negatively curved manifold

Let $M$ be a complete, noncompact Riemannian manifold with finite volume and whose sectional curvatures vary within the interval $[a,b]$, $-1\leq a<b<0$. It is known that such manifold has ends ...
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45 views

Locally exact vs globally exact

Why the volume form in Sphere is locally exact but not globally exact? here the integral is integral $$\int_{S^n}w$$ with $$w = \frac{1}{r} \sum_{i=1}^{n+1} (-1)^{i-1} x_i dx_1 \cdots\hat{dx_i},\cdots ...
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Proof a $(2n-1)$-compact manifold

I have no idea how prove that $$\{(z_0,\ldots,z_n)\in\mathbb{C}^{n+1} \quad| \quad z_0^d+z_1^2\ldots+z_n^2=0, \quad |z_0|^2+|z_1|^2\ldots+|z_n|^2=2\}$$ is a $(2n-1)$-compact manifold. How give the ...
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How to prove that $T\mathbb{S}^3 \to \mathbb{S}^3$ is a trivial 3-vector bundle? [closed]

How to prove that $T\mathbb{S}^3 \to \mathbb{S}^3$ is a trivial 3-vector bundle?