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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Existence of vector extensions for the Hessian

Q: $p$ a critical point of a smooth $f$ and $v, w$ two vectors in $T_{p}M$. We extend these two to vector fields $v^{*}, w^{*}$ such that at $p$ the first one equals $v$ and the second one equals $w.$ ...
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966 views

Volume form on a sphere.

Let $S^n(r)$ be the sphere of radius $r$ , $x_1^2 + ... + x_n^2 = r^2$ and let $$w = \frac{1}{r} \sum_{i=1}^{n+1} (-1)^{i-1} x_i dx_1 \cdots\hat{dx_i},\cdots dx_{n+1} $$ Write $S^n$ for the unit ...
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132 views

Existence of coordinate systems for submanifolds

I decided to do the following problem as an exercise: Let $p \in M$ be a regular point of $f: M \to \mathbb{R}$. Prove the existence of a coordinate system $(x_1,x_2,...,x_n)$ near $p$ such that ...
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132 views

[ANSWERED]Lie brackets on vector fields

We consider $v = \frac{\partial}{\partial x}$ and $w = x * \frac{\partial}{\partial z} + \frac{\partial}{\partial y}$. I need to first find the Lie bracket between them which i get to be: ...
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47 views

levels curves of polynomial equations as manifolds

Q: For which real values c is the subset $f(x) = x_{1}^{2} + x_{1}^{3} - x_{2}^{2} + x_{3}x_{4} = c$ a smooth submanifold of $\mathbb{R}^4$? Try: for it to be a smooth submanifold, $c$ has to be a ...
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230 views

Continuous function approximation on manifolds

I am asked to show that every cont. function from a manifold M to $\mathbb{R}$ can be approximated by smooth functions. Try: let f be a map from M to the reals(R). Let ${s_{i}, U_{i}}$ be our atlas. ...
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132 views

Whitney Embedding theorem for manifold with boundary in Lee's Introduction to smooth manifolds(2nd Edition)

I am reading through the new edition of Lee's book and I am stuck by the proof of Theorem 6.15. When passing to a non-compact manifold, the author begin by defining several sub-levelsets and claim ...
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66 views

how to obtain a generalized Morse function out of a fiber bundle?

Let $M\to E\to B$ be a smooth fiber bundle. In "Parametrized Morse Theory and Its Applications,(Proceedings of the ICM, 1990)", K. Igusa says that if dim $B$$<$dim $M$, then, there exists a smooth ...
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162 views

a condition for smooth vector field

Let $M$ be a Hausdorff manifold. I'm trying to prove that a vector field $Y:M\to TM$ is smooth if and only if the derivation induced by $Y$ for all globablly defined smooth functions is smooth. That ...
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80 views

A Milnor Differential Topology Excercise

If $m<p$, show that every map $f:M^m\longrightarrow\ S^p$ is homotopic to a constant, where $M^m$ is smooth manifold of dimension $m$. I tried to show that $M^m$ is contractible or convex, but I ...
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149 views

Why are normal bundles always locally trivial?

Is there a quick and dirty proof that normal bundles (say of some submanifold in a smooth manifold) are always locally trivial? My notes seem to have swept this assumption under the rug. Even ...
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48 views

Moving a compact submanifold off of another submanifold?

This is an intuitive idea that I see referenced a lot. Consider the following situation. Let $M$ and $N$ be submanifolds, $M$ compact, in some larger manifold $X$. Suppose also that ...
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1answer
74 views

Baire Category Theorem in a Smooth Manifold

Let $Z\subset M$ be a set of measure-0 , in a [smooth] manifold $M$. How does one shows that $M$ \ $Z$ is everywhere dense in $M$, using Baire category theorem? and which of the theorem version is ...
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36 views

a question in differential topology regarding signatures and Euler characteristics

Can you find an oriented, smooth, closed 4-manifold such that its Euler-characteristic is less than 3/2 times the absolute value of its signature and such that the signature isn't zero? The existence ...
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45 views

Continuity of point wise orientation

Prove that $[(X_1,...,X_n)]$ on a manifold $M$ is continuous if and only if every point $p$ in $M$ has a coordinate neighborhood $(U,\phi) = (U,x^1,...,x^n)$ such that for all $ q \in U$, the ...
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81 views

Analytic subvariety in complex manifold

I am trying to figure out a statement in a textbook "If $M$ is any complex manifold of a projective space $\mathbb{P}^{n}$, $V\subseteq M$ an analytic subvariety of dimension $k$, then we can find a ...
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54 views

$h$-principle for isometric embeddings

All the references I have seen so far list the Nash $C^1$-embedding theorem as an example where the $h$-principle holds. The $h$-principle for a differential relation holds by definition, when the ...
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130 views

Trivialisation of Möbius strip

I've just started studying Advanced Geometry and I'm in trouble with a (stupid) exercise. It's about finding a trivialisation of the Möbius strip (I'll refer to it as $E $) viewed as a fibre bundle ...
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77 views

Constructing vector bundles from local covers and transitions functions

Let $M$ be a smooth manifold. Suppose we are given an open cover ${U_\alpha}$ of $M$ and for $\alpha,\beta$ ; a smooth map $\tau_{\alpha\beta}\colon U_\alpha \cap U_\beta \to GL(k; R)$ satisfying the ...
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1answer
64 views

The sheaf of smooth functions is soft

A sheaf $\mathcal{F}$ over some topological space $X$ is called soft if any section over any closed subset can be extended to a global section (i.e. over $X$). How to understand that the sheaf of ...
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187 views

How does a left group action on the fiber of a principal bundle induce a right action on the total space?

Suppose I define a "principal $G$-bundle" as follows: A principal $G$-bundle is a fiber bundle $F \to P \overset{\pi}{\to} X$ with a left group action of $G$ on $F$ that is free and transitive, ...
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113 views

Proof that a set $X \subset M$ is a Manifold

Let M be a manifold without boundary and let , $g:M\to \mathbb R$ have $0$ as a regular value. Than the set $X \subset M$ with $g(x) \ge 0$ is a smooth manifold with boundary equal to $g^{-1}(0)$. I ...
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73 views

Smooth function with equibounded family of derivatives

By $\mathcal{C}^{\infty}(\mathbb{R})$ we denote the space of smooth functions $\mathbb{R}\rightarrow \mathbb{R}$. Also, by $\mathrm{supp}(f)$ we denote the closure $\mathrm{Cl}(f^{-1}(\mathbb{R}_{\ne ...
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153 views

Why do Zoll metrics exist only on $S^2$ and $RP^2$?

Zoll metric on a Riemannian manifold is a metric for which all geodesics are closed and have the same period. For sure, a standart metric on the sphere $S^2$ has this property: all its geodesics are ...
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299 views

Tubular neighbourhood theorem

in your opinion is it possible to get the existence of a tubular neighborhood for a manifold M even if it not embeds smoothly (but only topologically) in some R^N? Thank you!
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54 views

The Hopf invarient with coefficients other than Z.

So generally one defines the Hopf invariant of a map $f: S^{2n-1} \to S^n$ as the coefficient $H(f)$ in $\alpha^2 = H(f) \beta$ where $\langle \alpha \rangle = H^n(C_f)$ and $\langle \beta \rangle = ...
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1answer
257 views

Surgery on trivial knots

I know a theorem that any closed orientable 3 manifold can be obtained from the sphere $S^3$ by surgery along a framed knot. I think I read or heard somewhere that as a surgery link, we can take ...
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1answer
265 views

Finding Reeb Vector Field Associated with a Contact Form

I would greatly appreciate it if you could help me with the following: I'm curious as to how to find the Reeb field $R_w$ associated to a specific contact form $w$; does one actually find $R_w$ as ...
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184 views

First Pontryagin class on real Grassmannian manifold?

I wonder if real Grassmannian manifold $SO(p+q)/SO(p) \times SO(q)$ have nontrivial first Pontryagin class? I only have physics background and know really little about characteristic class theory.
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640 views

Distinguishing the Cylinder from a “full-twist” Möbius strip

Playing around with the definition of a fiber bundle, I found that while a Möbius strip (with its usual "half-twist") is a nontrivial fiber bundle, it seems that a Möbius strip with a "full-twist" is ...
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97 views

Stokes' Theorem for the upper half space.

How can I prove Stokes' Theorem $\int_M dω = \int_{∂M} ω$ where $M = \mathbb{H}^ n$, the upper half space.
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236 views

Why is the priemage of a regular value of $F(x,v)=df_x(v)$ finite?

This comes up as part of a larger issue of showing a compact $n$-dimensional manifold can be immersed in $\mathbb{R}^{2n-1}$ except at finitely many points. Suppose $X$ is a compact, $n$-dimensional ...
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1answer
43 views

Prove a$T_0$ topological group is $T_1$

How to prove that a $T_0$ topological group is $T_1$. I am a beginner in topological group. Also I want some good reference.
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1answer
125 views

Compact manifolds can almost be immersed?

The Whitney Immersion theorem states that any $n$-dimensional manifold can be immersed in $\mathbb{R}^{2n}$. However, I seem to remember that if $X$ is a compact $n$-dim manifold, then $X$ can be ...
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3answers
105 views

What dimensions are possible for contours of smooth non-constant $\mathbb R^n\to\mathbb R$ functions?

While for $n=2$ it is pretty clear that the contours of a non-constant $f:\mathbb R^n\to\mathbb R$ are either extrema (and therefore points) or (the union of) 1-dimensional isolines, for $n=3$ I am ...
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48 views

Does the structure group of S^n homotopic to O(n+1)?

It is easy to show that Diff(S^1) is homotopic to O(2), but in the case of n bigger than 1 things become really complicated, I cannot see the conclusion directly.
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Why isn't the graph of $y=|x|$ a smooth manifold? [duplicate]

Consider the graph of $y=|x|$ from $-1<x<1$. Equip it with a single chart, the projection onto the $x$-axis. Is it now a smooth manifold? It seems like it shouldn't be smooth, but perhaps with ...
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664 views

What is the codimension of matrices of rank $r$ as a manifold?

I'm reading through G&P's Differential Topology book, but I hit a wall at the end of section 4. There is a result stating The set $X=\{A\in M_{m\times n}(\mathbb{R}):\mathrm{rk}(A)=r\}$ is a ...
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1answer
46 views

Rotating about the z-axis defines a smooth function on $S^{2}$

I want to show that rotating about the z-axis defines a smooth function on $S^{2}$. To do this I used the function: $f(x,y,z)=(x\cos(\theta)-y\sin(\theta),x\sin(\theta)+y\cos(\theta),z)$ where ...
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100 views

Moduli space of stable principal $SL(2, {\mathbb C})$-bundles on a compact Riemann surface.

Let $C$ be a compact Riemann surface with genus $2$. How can I describe the moduli space of stable principal $SL(2, {\mathbb C})$-bundles on $C$?
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94 views

What's wrong with my argument to compute the vector bundles on $S^1$?

I have read the solution in Vector bundles of rank $k$ with base $S_{1}$, and I am sure that answer is correct. But I have another approach and I don't know where did I make a mistake. So as circle is ...
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Maximal submanifold

I wonder if the notion of "maximal submanifold" exists or is relevant? I'm surprised because I found pretty much nothing about it on the web (after a quick search). The definition, which seems ...
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2answers
203 views

Is there any standard N-sphere that has non-trivial first Pontryagin class?

I am wondering if there is a standard $N$-sphere that has non-trivial first Pontryagin class on its tangent bundle $TS^n$and frame bundle $FS^n$? I know that only $S^4$ has non-trivial $H^4(S^n, R)$ ...
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142 views

The degree of every smooth map $\mathbb{R}^n \to \mathbb{R}^n$ is one…

Let $\varphi : M^n \to N^n$ be a proper smooth map between two connected smooth manifolds. Then $\varphi$ induces a linear map $\varphi^* : H_c^n(N) \simeq \mathbb{R} \to H_c^n(M) \simeq \mathbb{R}$ ...
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2answers
135 views

Extending diffeomorphism to disk

I am trying to prove the following If $f:S^1 \to S^1$ is a diffeomorphism it can be extended to a diffeomorphism $F: D^2 \to D^2$. But I can't seem to prove it. I proved it for homeomorphisms using ...
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49 views

does composition of maps is smooth and one map is smooth imply the other is also smooth?

If $f\circ g$ is smooth and $f$ is smooth, does it follow that $g$ is smooth? Note that I cannot simply take the inverse of $f$. Do I have to use implicit function theorem?
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386 views

Diffeomorphism and determinant of Jacobian

I don't remember where I read it and if I remember it correctly but does the following hold true? If $M,N$ are two (smooth?) surfaces and $f: M \to N$ is a homeomorphism such that $det(J_f)$ (the ...
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Products of homeomorphisms

I was wondering if there is a theorem like "If $f_i:X_i\to Y_i$ are homeomorphisms then $\prod_i f_i : \prod_i X_i \to \prod_i Y_i$ is a homeomorphism" for $I$ finite. What about $I = \mathbb N$? ...
3
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82 views

Locally Euclidean but not topological manifold

I'm having trouble solving one part of one of the initial exercises of the classic Boothby book "An Introduction to Differentiable Manifolds and Riemannian Geometry" (exercise I.3.1). To be more ...
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345 views

The derivative of the inclusion map is the inclusion map of tangent spaces.

Let $X$ and $Y$ be smooth manifolds, let $i:X\to Y$ be the inclusion map, prove $di_x$ is the inclusion map from $T_x(X)$ to $T_x(Y)$. I know this is pretty basic, but can someone show me how to do ...