171 votes
Accepted

A continuous, nowhere differentiable but invertible function?

Interestingly, there are no such examples! For a continuous function $f : \mathbb{R} \rightarrow \mathbb{R}$ to be invertible, it must be either monotone increasing or decreasing. A famous classical ...
Alex Nolte's user avatar
  • 4,986
87 votes

If a two variable smooth function has two global minima, will it necessarily have a third critical point?

With respect to the first part of your question: No, a function with two global minima does not necessarily have an additional critical point. A counterexample is $$ f(x, y) = (x^2-1)^2 + (e^y - x^2)^...
Martin R's user avatar
  • 114k
60 votes

A continuous, nowhere differentiable but invertible function?

Invertible implies bijective by set theory, and bijective together with continuity implies strictly increasing or decreasing, which imply differentiability almost everywhere! (This is known as ...
A. Thomas Yerger's user avatar
53 votes
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Are diffeomorphic smooth manifolds truly equivalent?

The counterexample just shows that two diffeomorphic smooth structures on the same set $X$ do not need to share a common atlas. However, in any case, two diffeomorphic structures cannot be ...
Francesco Polizzi's user avatar
53 votes
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Why should I care about Gauss-Bonnet (and Gaussian curvature)?

In retrospect, this post got quite long. Also, the level varies greatly - sorry! Feel free to ask any questions. I guess I am quite fond of this topic, even though I am not as knowledgeable on it as ...
Alekos Robotis's user avatar
41 votes

Why is important for a manifold to have countable basis?

There is one point that is mentioned in passing in Moishe Cohen's nice answer that deserves a bit of elaboration, which is that a lot of the time it is not important for a manifold to have a countable ...
Eric Wofsey's user avatar
37 votes

If a topological space is homeomorphic to a smooth manifold, then will it be a smooth manifold?

Let me emphasize something that is perhaps not readily apparent in the other answers. "Smooth" is not something that a topological space is. You can't formally say "this topological space is a smooth ...
Najib Idrissi's user avatar
36 votes
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Can every continuous function between topological manifolds be turned into a differentiable map?

There is not necessarily a way to make a map smooth. For example, suppose $M=\mathbb{R}$, and suppose $N$ is any topological manifold of dimension $2$ or more. Let $f:M\rightarrow N$ be any ...
Jason DeVito - on hiatus's user avatar
34 votes

Coordinate-free definition of integration of differential forms?

There is a coordinate free approach to integration in the book Global Calculus by Ramanan, Chapter 3. In particular, the change of variable formula is deduced from abstract nonsense in Corollary 2.9. ...
user1066309's user avatar
34 votes

Are diffeomorphic smooth manifolds truly equivalent?

This is nothing specific to differentiable manifolds. If you have some set $X$ and you define some structure on $X$ (e.g. group, vector space, topological space, differentiable manifold, ...) there ...
M. Winter's user avatar
  • 29.9k
32 votes

If a two variable smooth function has two global minima, will it necessarily have a third critical point?

$ \def\norm#1{\lVert#1\rVert} $The answer to the question as stated is no as Martin showed, but is yes if we add the condition that $f(x)→∞$ as $\norm{x}→∞$. Martin's example pushes the saddle point '...
user21820's user avatar
  • 58k
31 votes

Where to start learning Differential Geometry/Differential Topology?

It's been around 11 months since I first asked this question. I thought I would share my path to learning Differential Topology and Differential Geometry. Hopefully this will be of some help to others ...
Perturbative's user avatar
30 votes
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Understanding the orientable double cover

Almost $2$ years later, I'll give a complete answer to my own question. Step 1 (Topology of $\widetilde{M}$): Take an atlas $\{(U_\alpha,\varphi_\alpha)\}_{\alpha\in\Lambda}$ such that $\{U_\alpha\}_{\...
rmdmc89's user avatar
  • 10k
29 votes
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Number of Differentiable Structures on a Smooth Manifold

The distinction to be made is that a differentiable structure is a choice of maximal smooth atlas $\mathcal A$, but two different choices $\mathcal A$ and $\mathcal A'$ can lead to isomorphic smooth ...
Pedro's user avatar
  • 122k
28 votes

When does a space admit a flat metric?

The octagon with edges identified appears to be different because you cannot tile the entire flat (Euclidean) plane with octagons in such a way that everything is OK when you cross the "border" and ...
Jeppe Stig Nielsen's user avatar
26 votes
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Why is important for a manifold to have countable basis?

Do you like to have a partition of unity? (For instance, in order to integrate differential forms.) Do you like your manifolds to embed in some ${\mathbb R}^N$? Admit a Riemannian metric? Do you like ...
Moishe Kohan's user avatar
26 votes
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From $e^n$ to $e^x$

We can show that if $$ f^{(n)} \ge 0\qquad{\rm(1)} $$ everywhere, for each integer $n >0$, and $f(x)=e^x$ on at least four points of $\mathbb R$, then $f(x)=e^x$ everywhere. This will make use of ...
George Lowther's user avatar
25 votes

Differential of the multiplication and inverse maps on a Lie group

One of Tu's previous exercises, ex. $8.7^*$ shows that for $M,N$ manifolds, by defining $\pi_1:M\times N\to M$ and $\pi_2: M\times N \to N$, one can show for $(p,q)\in M\times N$ that $$\pi_{1*}\times\...
snulty's user avatar
  • 4,365
25 votes
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Is the empty function differentiable?

Yes, the empty set is a smooth manifold (it is covered by the empty collection of coordinate charts!). It has every dimension. (That is, for any $n$, it is true that $\emptyset$ is a manifold of ...
Eric Wofsey's user avatar
23 votes
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Classifying space $B$SU(n)

In general for a group $G$, there exists a contractible free $G$-space $EG$ whose quotient $BG=EG/G$ is a classifying space for $G$. The construction of $EG$ is not unique and may be carried out ...
Tyrone's user avatar
  • 16.2k
23 votes
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Is the differential forms perspective on $dx$ incompatible with the technique of implicit differentiation?

What you're missing is that this computation does not take place in $\mathbb{R}^2$: it takes place in the submanifold $M$ of $\mathbb{R}^2$ defined by the equation $x^2+y^2=5^2$! We can consider $x$ ...
Eric Wofsey's user avatar
22 votes
Accepted

Why are geodesically convex sets diffeomorphic to $\Bbb R^n$?

We provide a careful proof of Theorem 5.1 from Bott & Tu. The main body of the proof is taken from this MO post. Let $M^n$ be a smooth manifold. An open cover $\{U_\alpha\}$ is good if each ...
Ryan Unger's user avatar
  • 3,516
22 votes
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When does a space admit a flat metric?

Let $M$ be a manifold and $G$ a discrete group acting freely and proper discontinuously, then $M/G$ is a manifold. If $g$ is a Riemannian metric on $M$, then it descends to a Riemannian metric on $M/G$...
Michael Albanese's user avatar
22 votes
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Reference for: "Surjectivity is just injectivity, only one dimension higher."

It's the other way around: injectivity is surjectivity but one dimension higher. This is a concept that comes up in a variety of homotopy-theoretic arguments. Here is a typical example. Theorem: Let ...
Eric Wofsey's user avatar
21 votes
Accepted

Where to start learning Differential Geometry/Differential Topology?

Differential Geometry by Barrett O'Neil and Introduction to Manifolds by Tu. The second is my all time favorite. It filled so many gaps for me.
Faraad Armwood's user avatar
21 votes
Accepted

Why aren't tangent spaces simply defined as vector spaces with same dimension as the manifold?

We don't just want to have a vector space to call the "tangent space". We want to do geometric things with the tangent space, and we can't do those things if it's just an arbitrary vector space of ...
Eric Wofsey's user avatar
20 votes
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Understanding Takens' Embedding theorem

Practical meaning of Takens’ Theorem using your example The butterlfly-like structure traced out by the trajectories of the Lorenz system is the attractor of this dynamics. Its properties contain ...
Wrzlprmft's user avatar
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20 votes
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Why is $\pi_7(\mathbb S^4)=\mathbb Z \oplus \mathbb Z_{12}$?

Start with the quaternion Hopf fibration $$S^3\xrightarrow{i} S^7\xrightarrow{\nu}S^4.$$ The map $\nu$ is an element of Hopf invariant one so generates the infinite cyclic summand in $\pi_7S^4$. The ...
Tyrone's user avatar
  • 16.2k
19 votes

A continuous, nowhere differentiable but invertible function?

If $f:(a,b)\to\Bbb R$ is continuous and injective it must be monotone, hence differentiable almost everywhere.
David C. Ullrich's user avatar
18 votes

Can we recover a compact smooth manifold from its ring of smooth functions?

[I assume all "smooth manifolds" are Hausdorff and paracompact.] Yes, you can recover $M$ as a smooth manifold from the ring $C^\infty(M)$. Here's a quick sketch. First, note that we can recover ...
Eric Wofsey's user avatar

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