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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Does $\Bbb{CP}^{2n} \# \Bbb{CP}^{2n}$ ever support an almost complex structure?

$\Bbb{CP}^{2n+1} \# \Bbb{CP}^{2n+1}$ supports a complex structure: $\Bbb{CP}^{2n+1}$ has an orientation-reversing diffeomorphism (complex conjugation!), so this is diffeomorphic to the blowup of ...
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22 views

Generalization of Inverse Function Theorem to noncompact submanifolds

In Guillemin and Pollack's Differential Topology, they give as an exercise (#1.8.14) to prove the following generalization of the Inverse Function Theorem: Use a partition-of-unity technique to ...
1
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1answer
15 views

Show that the outward unit normal is smooth vector field

Exercise 2.1.8 of Guillemin and Pollack asks us to prove that if $X^m \subset \mathbb{R}^n$ is an embedded submanifold with boundary, then the outward unit normal to $\partial X$ is a smooth function ...
5
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2answers
61 views

Is the connected sum of complex manifolds also complex?

Let $M$ and $N$ be real manifolds of dimension $n$ which happen to admit complex structures (so that necessarily $n=2k$ and both are orientable). Then does their connected sum $M\# N$ also admit a ...
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20 views

C^1 mapping of a non-metric topological space - does this make sense?

Is there a way to define a derivative on a mapping between general topological spaces without invoking a metric?
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1answer
50 views

Proof of Brouwer fixed point theorem using Stokes's theorem

$\omega$ is the volume form on the boundary B -ball $f\colon B \to \partial B$ $$ 0 < \int_{\partial B}\omega = \int_{\partial B} f^*(\omega) = \int_{B} df^*(\omega) = \int_B f^*(d\omega) = ...
2
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24 views

Characterization of graphs of maps between smooth manifolds

Theorem 6.52 in Lee's Introduction to Smooth Manifolds, 2nd ed., says Suppose $M$ and $N$ are smoothe manifolds and $S \subset M \times N$ is an immersed submanifold. Let $\pi_M$ and $\pi_N$ ...
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1answer
35 views

Proving a given set is a submanifold

Let $S \subseteq \mathbb R^n$. I have been faced with showing that $S$ is a submanifold and I have some ideas but I want to get the complete picture. (Main) Question 1: What methods are there to ...
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46 views

Are these “Stiefel-Whitney numbers” invariants of the (smooth)topology of the total space?

Let $E$ be a smooth real vector bundle over a smooth compact $n$-dimensional manifold $M$. It is relatively easy to see that the Stiefel-Whitney classes $w_i(E)$ are not diffeomorphism invariants of ...
2
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1answer
36 views

Locally flat manifold from Frobenius, differential forms approach

It is a known fact that a Manifold $M$ is locally flat if and only if the Riemann curvature tensor (Levi-Civita connection) vanishes. To show the if part is easy, but I am trying to follow the details ...
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37 views

Orientability and volume form

I was wondering if there is an easy argument that manifolds that are orientable always have a volume form and vice versa? So I am not looking for a full proof of this, but rather a good argument how ...
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28 views

Every differentiable structure is smoothable to a smooth structure.

I was studying differentiable manifolds and smooth manifolds. While reading on the Wikipedia website about them, I came across one statement that I have no idea why is it. This statement is Every ...
2
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1answer
43 views

Non-orientable submanifolds

Let $M$ be a $n$-manifold and let $S \subset M$ be a non-orientable $n$-dimensional submanifold possibly with boundary. Under what conditions can I conclude that $M$ is also non-orientable? Is ...
0
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0answers
31 views

Integrate symplectic two-form

I am supposed to show the following (maybe falsely stated) theorem Let $\Phi$ be a symplectic group action $\Phi: G \times M \rightarrow M$ on a manifold $M$. Assume that the symplectic form $\omega$ ...
5
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1answer
75 views

Diffeomorphism group of a manifold is never a Lie group?

Let $M$ be a smooth manifold. I heard there is a way to introduce a topology and a structure of infinite dimensional manifold (something like a Banach or a Frechet manifold) on $\text{diff}(M)$ ...
3
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74 views

Equivalent condition for non-orientability of a manifold

I've just came across this question, which gives us a great tool for showing that smooth manifold is non-orientable. Namely Thm. If $M$ is a smooth manifold and there are two charts ...
1
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1answer
37 views

Generalizations of Inverse Function Theorem

A beautiful exercise in Guilleman and Pollack asks us to show the following generalization of the Inverse Function Theorem: Suppose $f: M \to N$ is a map of smooth manifolds, and $Z$ is a ...
0
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1answer
35 views

Can a single point in a manifold be seen as a sub manifold?

In Pollack's differential topology, in Transversality, p.28, it reduced the study of the submanifold $Z$ to the simpler case, where $Z$ is a single point. But by the definition of manifold, it seems ...
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37 views

Reference request: Pullbacks along submersions are submanifolds and the induced map is a submersion

I'm looking for an introductory book in differential topology in which there are proofs of the following facts: Let $X,X',Y$ be smooth manifolds, $X\rightarrow Y$ a smooth map and $X'\rightarrow Y$ ...
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1answer
28 views

Preimage Lemma for transverse map. Help with some passages

I'm on my way proving the Preimage Lemma for a transverse smooth map but I've encountered some problems with two passages: Let $f\colon M \to N$ be a smooth map transverse to a submanifold $L$ of ...
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0answers
16 views

Is the restriction of a maximal atlas on an open submanifold maximal?

Let $M$ be a $n$-manifold, with some maximal atlas $A$, and let $V \subset M$ be an open set. The standard open-submanifold-atlas on $V$ is $A|_V$ defined as $$A|_V = \big\{ (U \cap V,x|_{U \cap V}) ...
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1answer
26 views

How does an atlas give a notion of whether a function is differentiable or not?

Defintion. An atlas is a set of pairs $\{(O_i,ψ_i)\}$ such that $\cup_iO_i=M$ and $\phi:O_i \rightarrow \mathbb{R}^n$, and most importantly, for $O_i \cap O_j \neq \emptyset $, $\tau_{ij}: \phi_i(O_i ...
0
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0answers
46 views

I have no idea what “smooth structure” is

I know what a manifold is: it's a topological space such that for every point there is an open set that looks like $\mathbb{R}^n$. But I do not know what a smooth manifold is, because I have no clue ...
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1answer
42 views

Is it (not) possible for two vector fields on the Klein bottle to be a basis?

Lefschetz fixed point theorem (for example) implies that a compact manifold with a nowhere vanishing vector field must have Euler characteristic zero. Is there a way to draw stronger algebraic ...
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15 views

Surfaces obtained by $\gamma$-reduction

$\mathcal{C}$ will denote the collection of all connected compact (not necessarily orientable) smooth 2-dimensional surfaces-without-boundary embedded in $M$ ( here $M$ is a complete Riemannian ...
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1answer
80 views

Is the sphere $S^2$ diffeomorphic to a quotient of the square?

If we take the square $[0,1]\times [0,1]$ and collapse the border, the resulting quotient space is homeomorphic to the sphere. The same holds if we take the square $[0,1]\times [0,1]$ with the ...
3
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2answers
32 views

2-form whose self-wedge does not vanish?

I know that any 2-form is decomposable if and only if its self-wedge vanishes. Is there an element $β ∈ A_2(R^n)$ such that $β ∧ β \neq 0$. Obviously, this $\beta $ must be indecomposable, but I ...
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0answers
40 views

Integral curves on non compact manifolds

Define a vector field on $\mathbb{R}^d$ by $X = \frac{\partial}{\partial x_{d}}$. That is a vector field that always points upward along the $x_{d}$-axis. Consider starting at any point $p \in ...
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1answer
101 views

Parallelizability of Lie groups

I have tried a lot to prove this well-known result. The basic idea behind the proof is clear to me. But I'm stuck at showing either of the following conditions: The map $\ G \rightarrow TG \ $ given ...
4
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2answers
65 views

Can $S^4$ be the cotangent bundle of a manifold?

I am asking the question because in the classical mechanics book by Arnold, he states that there is a distinguished $1$-form on $T^*V$. It seems that there is no such distinguished $1$-form on a ...
6
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2answers
51 views

Do two embeddings of a Euclidean space into a higher dimensional one only differ by a diffeomorphism?

Let $d\le n$ and $$f,g\colon\mathbb{R}^d\hookrightarrow\mathbb{R}^n$$ be two smooth embeddings. Is there a diffeomorphism $$\phi\colon\mathbb{R}^n\rightarrow \mathbb{R}^n,$$ such that $$f=\phi\circ ...
0
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1answer
15 views

Existence of solution to second order linear PDE

Suppose $f$ is a given smooth function on $\mathbb{R}^2$. I want to show that for $a,b,c \in \mathbb{R}$ such that $b^2 - ac > 0$ there exists a smooth function $u$ such that $$ a\frac{\partial^2 ...
0
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2answers
95 views

Prerequisites for differential topology

I want to self study differential topology. I know the basics of point set topology. I need some multivariate calculus and linear algebra, what is a good and short set of books which covers what I ...
2
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1answer
32 views

“Winding number”, Chern character and relative signatures of the metric

Anyone answer with good explanation is appreciated. In differential geometry, we discuss about topological quantities like characteristic classes. For example, the first Chern character of some ...
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0answers
20 views

Existence of solution to linear second order PDE.

Suppose $f$ is a given smooth function on $\mathbb{R}^2$. I want to show that for $a,b,c \in \mathbb{R}$ such that $b^2 - ac > 0$ there exists a smooth function $u$ such that $$ a\frac{\partial^2 ...
1
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0answers
34 views

Show that $1$-form has particular coordinate representation.

I want to show that for a nowhere-vanishing $1$-form $\alpha$ on an $n$-dimensional manifold $M$ we have that $\alpha \wedge d\alpha = 0$ if and only if around every point $p$ there exists a ...
0
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0answers
20 views

If $X$ is a $k$-dimensional manifold with boundary, then $\partial X$ is a $(k-1)$ dimensional manifold without boundary.

Definition : (1) A subset $X$ of $\mathbb{R}^N$ is a $k$-dimensional manifold with boundary if every point of $X$ possesses a neighborhood diffeomorphic to an open set in the space $\mathbb{H}^k$. ...
2
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1answer
24 views

Showing $\mathbb R^n$ is a smooth premanifold

I am attempting to fill in a proof that $\mathbb R^n$ is a smooth premanifold and its smooth functions are what one would expect: the infinitely differentiable functions from $\mathbb R^n$ to $\mathbb ...
0
votes
1answer
70 views

Show that inf $f(x)$ is achieved. Find $\inf f(x)$.

Let $$\Sigma = \{x\in R^3: x_1x_2 +x_1x_3 +x_2x_3=1 \}$$ and $$f(x) = x_1^2 + x_2^2 + \frac{9}{2} x_3^2$$ a) Show that $\Sigma$ is a smooth surface in $R^3$. b) Show that $\inf_{x\in\Sigma}$ f(x) ...
3
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1answer
41 views

Perpendicular Gradients

Suppose $f:\mathbb{R}^2\to \mathbb{R}$ is smooth. Further suppose $\nabla f$ vanishes no where. When is it possible to find a smooth non-singular $g:\mathbb{R}^2\to \mathbb{R}$ satisfying $\nabla ...
3
votes
1answer
184 views

Lie bracket; confusing proof from lecture

I am having some difficulties understanding this proof. Let $G$ be a closed matrixsubgroup of the general linear group. We have a right translation $Y(g):=dR_g(e) Y(e)$ on the Lie algebra $Y \in ...
2
votes
2answers
83 views

What is topologically the set of all straight lines in $\mathbb{R}^d$? More structures on it?

If we consider the set of all straight lines in $\mathbb{R}^d$, then what is it topologically? If it's topologically 'nice' i.e. a manifold, probably we could put a smooth manifold structure on it ...
3
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1answer
51 views

Extending a function on embedded submanifold: what techniques are to be used? (2.3, Lee's Riemannian Manifolds)

I am asking about a slightly different version of this question, where we are given an embedded submanifold $M \subset M'$ and are asked to extend any smooth function on $M$ to one on a neighborhood ...
2
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1answer
32 views

Without loss of generality in proof about subspaces in symplectic linear algebra

A linear symplectic space is a 2n-dimensional vector space $V$ with a symplectic two form $\omega.$ On this vector space $V$ is a canonical basis $(e_1,...,e_n,f_1,...f_n)$ with $\omega(e_i,f_j) = ...
0
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1answer
22 views

Tangent spaces and derivatives are defined in the setting of manifolds with boundary

First, suppose that $g$ is a smooth map of an open set $U$ of $\mathbb{H}^k$ into $\mathbb{R}^l$. If $u \in \partial U$, the smoothness of $g$ means that it may be extended to a smooth map $\phi$ ...
3
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1answer
47 views

Does the fixed point in the Brouwer's Fixed-Point Theorem have to be an interior point?

As the question suggests, does the fixed point in the Brouwer's Fixed-Point Theorem have to be an interior point? Many thanks in advance. To be clear, I'm using the statement of Brouwer's Fixed-Point ...
5
votes
3answers
74 views

Compute $\int_M \omega$

Let $M=\{(x,y,z): z=x^2+y^2, z<1\}$ be a smooth 2-manifold in $\Bbb{R}^3$. Let $\omega=xdy\wedge dz+ydz\wedge dx+zdx\wedge dy\in \Omega^2(\Bbb{R}^3)$. Compute $$\int_M \omega.$$ I parametrised ...
5
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0answers
55 views

Manifold, exist smooth nonnegative function with regular value at 0?

If $X$ is any manifold with boundary, then does there exist a smooth nonnegative function $f$ on $X$, with a regular value at $0$, such that $\partial X = f^{-1}(0)$?
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1answer
27 views

Quotient group and kernel of canonical projection

Imagine we have a group $G$ acting properly and freely (as a group action $\Phi: G \times M \rightarrow M$) on a manifold $M$, then $M/G$ is a manifold and there is a smooth submersion $\pi: M ...
4
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0answers
109 views

Index of zero of a gradient vector field at a critical point

Let $M$ be a Riemannian manifold with a Morse function $f: M \to \mathbb{R}$. The zeroes of the gradient vector field of $f$ are the critical points of $f$. How do you show that a critical point ...