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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Is an immersed submanifold second-countable?

By general manifold I mean Hausdorff differential manifold not necessarily second-countable. By standard manifold I mean Hausdorff, second-countable differential manifold. So my question is, we have ...
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630 views

On the Use of the Topology on Tangent Bundles

On learning about how to define smooth vector fields on a manifold $M$, I learned that one should first define a tangent bundle , $T(M)$, as $\cup T_p(M)$ together with a topology(smooth structure). ...
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115 views

On Tangent vectors as jets & submanifolds

Here is my second question on understanding jets better: For a smooth manifold $M$ the set of jets $J^1_0(\mathbb{R},M)$ is the same as the Tangent bundle $TM$. This implies that any equivalence ...
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1answer
90 views

Is there any connection between partial derivative and matrices?

I can see in some texts and books that the authors use big letters in order to describe partial derivative of function in $\mathbb{R^n}$ similar to the way we write matrices in linear algebra, for ...
5
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1answer
266 views

Morse theory and surfaces

I have heard that one can prove the classical classification of surfaces theorem using Morse theory. I am planning on learning this approach as a way to motivate and get comfortable with Morse ...
0
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1answer
165 views

Let $\mathbb{R}^m_+ = \lbrace x \in \mathbb{R}^m : x \geqslant 0 \rbrace.$ What is the boundary set of the set $ \mathbb{R}^m_+$?

Let $ x \in \mathbb{R}^m,$ $ x = [x_i], i = 1,2, \dotsm , m.$ Define $\mathbb{R}^m_+ = \lbrace x \in \mathbb{R}^m : x_i \geqslant 0, 1 \leqslant i \leqslant m \rbrace.$ What the boundary set $ ...
7
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393 views

Stokes for integration along the fiber

I want to use a version of Stokes theorem for integration along the fiber and I need some help in proving a general statement. Let $F$ be a $k$-manifold with boundary and let $E \to M$ be a smooth ...
3
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1answer
273 views

Knot with genus $1$ and trivial Alexander polynomial?

I would like to know whether there exists a knot $K$ with genus $g(K)=1$ and trivial Alexander polynomial $\Delta(K) \doteq 1$. A linked question could be: does there exist a Whitehead double with ...
4
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2answers
164 views

Why is the pullback completely determined by $d f^\ast = f^\ast d$ in de Rham cohomology?

Fix a smooth map $f : \mathbb{R}^m \rightarrow \mathbb{R}^n$. Clearly this induces a pullback $f^\ast : C^\infty(\mathbb{R}^n) \rightarrow C^\infty(\mathbb{R}^m)$. Since $C^\infty(\mathbb{R}^n) = ...
13
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1answer
210 views

What's so special about a homotopy $15$-sphere?

I just saw a table which counts diffeomorphism classes of homotopy $n$-spheres (that is, spaces homotopy equivalent to $n$-dimensional spheres). Such a table can be seen on the first page of this ...
0
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1answer
211 views

Preimage of submanifold under an embedding

Suppose we have two smooth manifolds $M_1$ and $M_2$ and a smooth map $i:M_1 \rightarrow M_2$ that is an embedding of $M_1$ into $M_2$. Moreover we have another submanifold $N \subset M_2$ that has a ...
4
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1answer
167 views

Image of smooth vector bundle morphism

Let $\pi_1:V_1 \rightarrow B_1$ and $\pi_2:V_2 \rightarrow B_2$ be smooth vector bundles and write $\phi_V: V_1 \rightarrow V_2$ as well as $\phi_B:B_1 \rightarrow B_2$ respectively for the total and ...
2
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0answers
180 views

How does a geodesic equation on an n-manifold deal with singularities?

My general premise is that I want to investigate the transformations between two distinct sets of vertices on n-dimensional manifolds by: Minimalizing the change in the fundamental shape of the ...
4
votes
1answer
260 views

A theorem in Morse theory

If $M$ is a smooth manifold on which there is a smooth function $f:M \to ( - 1,2)$ such that all $[0,1]$ are regular values of $f$ and ${f^{ - 1}}(s)$ is a compact set for all $s \in [0,1]$,then is ...
5
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1answer
200 views

$\{\gamma \in C^0([0,1],M): \gamma (0)=p, \gamma (1)=q\}$ with the compact-open top has the homotopy type of a CW complex

...where $M$ is a smooth manifold and $p, q \in M$. Does anyone know of any slick or accessible proofs of this? I was referred to Milnor's "On Spaces Having the Homotopy Type of a CW Complex" which ...
2
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0answers
124 views

Action of U(1) on a sphere bundle

Suppose $N$ is a closed, $n$-dimensional, orientable smooth manifold. Moreover, $n$ is odd. Consider the tangent bundle $TN$ of $N$. By adding a point at infinity to each tangent space I define a ...
0
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1answer
99 views

Non-intersecting smooth paths in the plane and the relation to curvature

I'm interested in a problem and have no idea how to approach solving it. Could you please point me in the right direction. Given 3 smooth paths $\varphi_{1}:[0,1]\to \mathbb{R}^{2}$, ...
4
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2answers
599 views

What can Lefschetz fixed point theorem tell us about the group of automorphisms of a compact Riemann surface?

Let $X$ be a compact Riemann surface of genus $g>1$, $f\in Aut(X)$, a biholomorphism of $X$ onto itself, $x\in X$ a fixed point of $f$. Since tangent map of a holomorphic map (on the real tangent ...
25
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10answers
4k views

“Immediate” Applications of Differential Geometry

My professor asked us to find and make a list of things/facts from real life which have a differential geometry interpretation or justification. One example is this older question of mine. Another ...
9
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1answer
907 views

Diffeomorphism group of the unit circle

I am given to understand that the group of diffeomorphisms of the unit circle, $\operatorname{Diff}(\mathbb{S}^1)$, has two connected components, $\operatorname{Diff}^+(\mathbb{S}^1)$ and ...
2
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3answers
191 views

Real analytic diffeomorphisms of the disk

Is there any real analytic diffeomorphism from two dimensional disk to itself, except to the identity, such that whose restriction to the boundary is identity?
4
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0answers
93 views

extension to the ball of sphere immersion

What are the constraints to extend an immersion of the sphere $S^2$ into $\mathbb{R^3}$ to an immersion of the closed unit ball $B(0,1)$ to $\mathbb{R}^3$? Suppose, I get an immersion of $S^2$ into ...
4
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2answers
471 views

Properties preserved by diffeomorphisms but not by homeomorphisms

Diffeomorphisms between manifolds are particular homeomorphisms, so each property preserved by homeomorphisms is preserved by diffeomorphisms. Can you show me some examples of properties preserved by ...
4
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1answer
101 views

metric on the Euclidean Group

I am not an expert in this so I hope this doesn't sound so stupid: what is the common metric used when studying the Euclidean Group $\mathrm{E}(3)$. One could probably ask the same thing for ...
2
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1answer
165 views

Higher dimensional Jordan-Brower separation theorem

Is there an established higher dimensional version of the Jordan-Brower theorem? I would like some statement like this: Let $M$ be an $n$-dimensional closed, compact and connected variety and suppose ...
3
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1answer
620 views

Open subsets in a manifold as submanifold of the same dimension?

An open set in an $n$-manifold is clearly a submanifold of the same dimension as its containing manifold (see open manifolds). Now, given an $n$-manifold $M$, is it true that a set, to be the ...
2
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1answer
126 views

The embedding of manifold

If $M$ is a smooth manifold and $f$ is a smooth function on it, is it necessarily that there is an embedding $F$ of $M$ to $\mathbb R^k$ such that the first component of $F$ is $f$ ?
4
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1answer
144 views

Holomorphic immersion

Is it possible to get an holomorphic immersion form $\mathbb{C}$ to $\hat{\mathbb{C}}$ which is surjective? Here immersion means that $f'(z)\neq 0$ when $f(z)$ is finite and ${\left(\frac ...
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0answers
126 views

Is the Euler Number of A Vector Bundle Always Finite?

Let $E$ be an oriented vector bundle with an even rank $n$ over a smooth oriented $n$-dimensional manifold $M$, $e(E)$ denotes the corresponding Euler class, then the Euler number is defined by ...
0
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1answer
115 views

Can one average “close” smooth functions?

Suppose $M$ is a connected, smooth, second-countable manifold. Let $U \subset M^n$ be some neighbourhood of the diagonal. We will call a function $a: U \times \Delta_n \rightarrow M$ an "averaging ...
2
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1answer
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Poincaré Duality with de Rham Cohomology

This is probably a fairly basic question: Poincaré Duality states the following: Given an $n$-manifold, the $k^{th}$ homology is isomorphic to the $(n-k)^{th}$ cohomology. So I was curious is there ...
2
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1answer
454 views

Compact submanifold represents a homology class

I've heard somewhere that a compact submanifold of a manifold gives a homology class of the whole manifold and vice versa. Can somebody explain this to me?
3
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1answer
187 views

What's the shining sparkle in the Sphere Inside-Out problem?

I've just seen the wonderfully done movie Sphere Inside Out, one about the Smale's paradox. And the first question came in mind is that, why it has to be so ugly? Why turning an ultimately simple ...
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2answers
222 views

Diffeomorphisms of $\mathbb{R}^n$

I recently realized that I don't know any non-linear diffeomorphisms of the plane (or $\mathbb{R}^n$ in general) except for linear ones, so I want to ask rather broad questions hoping to be pointed to ...
2
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0answers
238 views

How to construct a vector field without zero on an open manifold?

a friend asked me to pose the following problem: It is known that on an open manifold (connected, not compact and without boundary) there exists a vector field without zero, since its Euler ...
3
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0answers
400 views

Ambient Isotopy

From Hirsch's Differential Topology, p. 180. The first of the isotopy extension theorems says; Let $A\subset M$ be a compact submanifold and $F:V\times I \rightarrow S^{3}$ an isotopy of $A$. If ...
14
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1answer
308 views

Question about the proof of the index theorem appearing in Milnor's Morse Theory

I am trying to get through the proof of the index theorem. The background: I have been stuck for quite a while on the following point which Milnor says is evident: Let $\gamma: [0,1]\rightarrow M$ be ...
2
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1answer
192 views

Locally flat submanifold

Recently I found the following definition: Let $M^{n}$ be an $n-$dimensional topological manifold. Then $N^{k}\subseteq M^{n}$ is a locally flat submanifold if for every $x\in N$ there exists an open ...
6
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1answer
274 views

Proof that vector area is a boundary integral?

Let $M \subset \mathbb{R}^2$ be a closed topological disk and let $f: M \rightarrow \mathbb{R}^3$ be a smooth embedding; let $N$ be the corresponding unit normal field on $M$. The vector area is ...
6
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1answer
234 views

Are all vector bundles “flat vector bundles”?

This concept appears in Bott&Tu's GTM82. A flat vector bundle is one who has a particular trivialization with locally constant transition functions. Then my question is whether every vector bundle ...
0
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1answer
105 views

equivalence between differentiability definitions

In analysis course we encounter commonly the following definition of differentiable function: $f:U \rightarrow \mathbb{R^m}$, where $U \subset \mathbb{R^n}$ is differentiable when $\exists \ T \in ...
6
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1answer
293 views

An example of a derivation at a point on a $C^k$-manifold which is not a tangent vector

Let $M$ be a real $C^k$-manifold, $\mathscr{H}_x$ be the $\mathbb{R}$-algebra of germs of $C^k$-differentiable functions at $x \in M$. It is easy to show that $T_x M$ can be embedded into the space of ...
6
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2answers
185 views

Is a vector bundle orientable if and only if its dual bundle is orientable?

I was reading up on my dual spaces today and I made the following hypothesis: A vector bundle $\xi$ is orientable if and only if $\xi^*$ is orientable. This seems rather intuitive, and although ...
10
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2answers
370 views

Reversing the Ricci flow

Suppose $S$ is a closed, oriented surface (2-manifold) embedded in $\mathbb{R}^3$, which inherits the metric from $\mathbb{R}^3$, so that distances are measured by shortest paths on the surface. If ...
6
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1answer
182 views

Obstructions to lifting a map for the Hopf fibration

This is a bit of an elementary question, but suppose $\pi: \mathbb{S}^3\to \mathbb{S}^2$ is the Hopf fibration, are there reasonably computable obstructions to when a map $f:M\to \mathbb{S}^2$ can be ...
2
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1answer
110 views

A question about complex manifolds

Let $(M,J_{M})$ be a almost complex manifold and $(N,J_{N})$ be a complex manifold. I want to prove that $F^{*}(\mathcal{O}_{N})\subset\mathcal{O}_{M}$ implies that $F:M\rightarrow N$ is almost ...
15
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1answer
1k views

What does “locally trivial” do for us?

For the following we will work in the smooth category. (But examples in the topological category is also welcome.) The usual definition of a fibre bundle is Def A fibre bundle is the quadruple ...
5
votes
2answers
487 views

basic differential forms

Given a fiber bundle $f: E\rightarrow M$ with connected fibers we call the image $f^*(\Omega^k(M))\subset \Omega^k(E)$ the subspace of basic forms. Clearly, for any vertical vector field $X$ on $E$ we ...
4
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2answers
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An exercise in Spivak's *Calculus on Manifolds*

Problem 5-6 in Michael Spivak's Calculus on Manifolds reads: If $f:\mathbb R^n\to\mathbb R^m$, the graph of $f$ is $\{(x,y):y=f(x)\}$. Show that the graph of $f$ is an $n$-dimensional manifold if ...
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1answer
194 views

Are there bounded surfaces without boundary that are noncompact?

I am aware of the Heine-Borel theorem, which says that closed and bounded is the same as compact in $\mathbb{R}^n$. My question is: are there (connected) surfaces without boundary embedded in ...