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.

learn more… | top users | synonyms

1
vote
2answers
67 views

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 ...
4
votes
2answers
77 views

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 ...
3
votes
1answer
91 views

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 ...
0
votes
0answers
31 views

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 ...
2
votes
1answer
34 views

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 ...
5
votes
1answer
65 views

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 ...
3
votes
1answer
36 views

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 ...
0
votes
0answers
23 views

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 ...
1
vote
1answer
38 views

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: ...
0
votes
0answers
17 views

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 ...
7
votes
2answers
249 views

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 ...
0
votes
0answers
13 views

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? ...
3
votes
1answer
42 views

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 ...
2
votes
1answer
25 views

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 ...
1
vote
1answer
31 views

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 ...
2
votes
1answer
33 views

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})} ...
1
vote
0answers
30 views

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) ...
1
vote
3answers
67 views

Degree 1 map from torus to sphere

I'm trying to find a smooth degree 1 map from the torus $T^2 = S^1 \times S^1$ to the 2-sphere $S^2$. My first thought was to use the two coordinates $(\theta_1,\theta_2)$ to map onto the usual ...
2
votes
2answers
79 views

de Rham cohomology on finitely smooth manifolds

In all of the places I've looked, de Rham cohomology is defined on $\mathcal{C}^\infty$ manifolds with $\mathcal{C}^\infty$ differential forms. What about de Rham cohomology on $\mathcal{C}^r$ ...
0
votes
1answer
23 views

A smooth function $f$, defined on an open ball in $\mathbb{R}^n$, can be written the sum of $n$ smooth functions with a certain property

Let $f: B \to \mathbb{R}$ be a $C^\infty$ function on an open ball $B := B_r(a) \subseteq \mathbb{R}^n$. I want to show that there exist $C^\infty$ functions $g_1, ..., g_n: B \to \mathbb{R}$ with ...
0
votes
0answers
26 views

Discretizations of Differential, Geometric and Topological Notions

I have noticed a recurring theme in Graph Theory / Theoretical Computer Science (abbreviated GT and TCS throughout this post) in that notions typically belonging to differential calculus / geometry / ...
4
votes
0answers
157 views

Homotopy of space of immersions, Smale-Hirsch theorem

If $M$ and $N$ are manifolds with $\dim M< \dim N$, we denote by $Imm\left(M,N\right)$ the space of immersions of $M$ in $N$. Let $M$ and $M'$ manifolds of dimensions $m>0$. It is true that if ...
1
vote
1answer
38 views

An orientation on a $(n-1)$-dimensional submanifold.

This question goes on where this question ended. Given is an non-empty $(n-1)$-dimensional ($n\ge2$) differentiable submanifold $X\subset\mathbb{R}^n$ such that there exists an open ...
2
votes
2answers
44 views

Topological boundary of a specific manifold.

Let $\emptyset\neq X\subset\mathbb{R}^n$, $n\ge2$, be an $(n-1)$-dimensional differentiable submanifold, i.e. for every $p\in X$ there is an open $U_p\subset\mathbb{R}^n$ with $p\in U_p$, and a ...
2
votes
1answer
66 views

Is $S^3\times S^2$ orientable?

The question comes from another question, I am asked to calculate the dimension and check orientability of the manifold $$ V_2(\mathbb{R}^4) = \{(v_1,v_2) \in \mathbb{R}^4\times \mathbb{R^4} \mid ...
1
vote
2answers
48 views

Why are tangent vectors coordinate-dependent?

Why does the coordinate basis for $T_pM$ depend on the coordinate chart we are using? Any two charts containing $p$ agree on some neighborhood of $p$, so shouldn't we be able to find a basis for ...
2
votes
1answer
40 views

Can we lower bound the volume of the image of a ball under a diffeomorphism?

Apologies if this question is overly simple, I'm new to differential geometry. Suppose I have two Riemannian manifolds $M_1$ and $M_2$, along with a diffeomorphism $f:M_1\to M_2$ between them. Let ...
4
votes
1answer
37 views

Finding all $2$-forms in the right half-plane that are invariant under glide transformations

I'm trying to find all 2-forms $\omega$ that are invariant under glide transformations in the right half-plane, $X = \{ (x,y) \in \mathbb{R}^2 : x > 0\}$. To do this, we can write the vector field ...
8
votes
2answers
158 views

Proving that $N$ is a manifold.

I'm dealing with the following exercise from Munkres' "Analysis on Manifolds": Let $f:\mathbb R^{n+k}\rightarrow \mathbb R^n$ be of class $C^r$. Let $M$ be the set of all $x$ such that $f(x)=0$. ...
2
votes
1answer
33 views

$A\cup \text{Fr}A$ is a manifold

I'm dealing with the following exercise from Spivak's "Calculus on Manifolds": Let $A\subset \mathbb R^n$ be an open set such that the (topological) boundary of $A$, $\text{Fr}A$ is an $n-1$ ...
1
vote
1answer
32 views

Every open cover of a smooth Manifold has a regular refinement

I am trying to understand the proof of Let M be a smooth manifold. Every open cover of M has a regular refinement. The proof begins as follows [Lee] : Let $X$ ...
0
votes
1answer
16 views

Is there exist a smooth function $f:\mathbb{R}^{m}\to M$ such that $f|_{U′}=\phi^{-1}$?

Let $M$ topological smooth manifold and $(U,\phi)$ chart fixed with $\phi(U)=U′$ open in $\mathbb{R}^{m}$. Is there exist a smooth function $f:\mathbb{R}^{m}\to M$ such that $f|_{U′}=\phi^{-1}$? I ...
1
vote
3answers
40 views

How these charts are written?

The spherical coordinate map$$σ(u, v) = (\cos u \cos v, \cos u \sin v,\sin u), −π/2 < u < π/2, −π < v < π,$$ and its variation $$σ˜(u, v) = (\cos u \cos v,\sin u, \cos u \sin v), −π/2 < ...
0
votes
0answers
20 views

Prove that integration of a differential $k$-form is independent of choice of basis

This is Exercise 4 of Section 33 of Munkres' "Analysis on Manifolds" book: (Let $A$ be an open set in $\mathbb{R}^k$.) If $\eta$ is a $k$-form in $\mathbb{R}^k$ and if ...
0
votes
1answer
23 views

Reference about the space of closed curves in Riemann manifold

Some days ago, I listened a report about the width of Riemann manifold. I am interesting in the space of closed curves in Riemann manifold. It seemly has good topology and different construction.For ...
2
votes
0answers
30 views

Smoothing a continuous section in a fibre bundle.

Let $\pi: X \rightarrow M$ be a smooth fibre bundle and let $p^{1}_{0} : X^{(1)} \rightarrow X$ be its 1-jet bundle. Suppose there is a $\mathcal{C}^{1}$ section $h: M \rightarrow X$ such that it is ...
1
vote
0answers
26 views

On lifts of a trajectory of a quadratic differential

Let $X$ be a Riemann surface of genus $g\ge 2$ and $q$ a holomorphic quadratic differential on $X$. The differential $q$ defines a flat metric with conical singularities on $X$: if $q=f(z)dz^2$ ...
1
vote
0answers
21 views

Cobordism Groups an the Pontryagin-Thom Construction

I am confused by the statement that $\Omega^\text{framed}_1(S^3) \cong \mathbb{Z}$ which I came across as an application of the Pontryagin-Thom construction for showing that $\pi_3(S^2) \cong ...
0
votes
0answers
14 views

Generate Lie-Algebra $su(N)$

Let $A \in \mathbb{C}^{n \times n}$ be a diagonal matrix with $Tr(A)=0$ and all eigenvalues (purely imaginary eigenvalues) are mutually different from each other. Hence, in particular $A \in ...
3
votes
1answer
39 views

Does there exist a non-parallelisable manifold with exactly $k$ linearly independent vector fields?

For any positive integer $k$, is there a smooth, closed, non-parallelisable manifold $M$ such that the maximum number of linearly independent vector fields on $M$ is $k$? Note that any such $M$, ...
0
votes
1answer
19 views

Computing the degree of a one-variable map

Edit: This question originally contained a typo where the function $f$ specified below was equal to $x$, not $x^2$ as currently written, outside an interval $[-T,T]$, and the accepted answer was ...
10
votes
3answers
88 views

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

It is well-known that if $X$ is a reasonably nice topological space (compact Hausdorff, say) then we can recover $X$ from the ring $C(X)$ of continuous functions $X\to\mathbb R$; see this MSE question ...
2
votes
1answer
38 views

Example of a diffeomorphism from all of $\mathbb{R}$ to itself

I can think of diffeomorphisms from an interval to $(a,b)\rightarrow \mathbb{R}$, scaling the tangent function, and from the punctured plane, polar coordinates, or some odd polynomial, but does anyone ...
0
votes
0answers
24 views

O.D.E. in Homogeneity Lemma

Let $\psi: \mathbb{R}^{n} \to \mathbb{R}$ smooth such that $\psi(x) > 0$ for $x \in B(0,1)$ and $\psi(x) = 0$ for $x \notin B(0,1)$. Let $c \in S^{n-1}$ fix and arbitrary and consider the O.D.E. ...
0
votes
0answers
47 views

Computing relative de Rham cohomology?

Let $f: N \to M$ be a smooth map. Then in Bott and Tu they define a relative de Rham complex $\Omega^{\bullet}(f)$ where $\Omega^k(f) = \Omega^k(M) \oplus \Omega^{k-1}(N)$ with coboundary map given in ...
0
votes
0answers
15 views

Which constructions on vector bundles satisfy a universal property?

I'm wondering whether constructions on vector bundles, such as the Whitney sum or tensor product of two bundles, satisfy some kind of universal property. What I mean by "some kind of universal ...
3
votes
0answers
29 views

How to keep symplecticity of a diffeomorphism after a coordinate rescaling and Taylor series expansion?

Let's take some, implicitly written, symplectic diffeomorphism $F$ of $\mathbb{R}^{2n}$, let it preserve the (possibly non-standard) symplectic form $\omega$. Suppose I introduce a small parameter ...
1
vote
0answers
15 views

Why do we need the $S \otimes L^{1/2}$ bundle product to determine a $Spin_c$ structure?

I am reading Marino's book on topological field theory and 4-manifolds and I am very confused in the construction of the $Spin_c$ structures for manifolds that do not admit a $Spin$ structure. In the ...
1
vote
0answers
21 views

Construct a lifting from the the total space of a fiber bundle to its spoace of 1-jets

I am reading the book "Convex Integrtion Theory by D.Spring" and need some help. Let $\pi:X \rightarrow V$ be a smooth fiber bundle over a smooth base manifold $V$. Let $h\in\Gamma^{0}(X)$ and ...
2
votes
2answers
65 views

What does $C^k$ at a single point mean?

I'm reading textbooks on manifolds. I usually see a saying that "function $f$ is $C^k$ at a point $p \in M$". I am confused with this. What does this mean? Furthermore if $f$ is $C^k$ at $p$, is it ...