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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Non-vanishing vector fields on non-compact manifolds

In several papers the following result is invoked: Theorem. Every connected, non-compact, smooth manifold $M$ carries non-vanishing smooth vector fields $v$. (we are assuming $M$ is $2$nd countable ...
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Topological spaces with prescribed fundamental groups

The question I am about to ask could have gone to the chat section but I want to have the answers/comments in an easy-to-refer-back-to style. For (connected, pointed) topological spaces with trivial ...
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171 views

Is the map from the circular half cone to the $xy$ plane a local isometry?

This is a text book exercise. And I think that this map is not a local isometry. But, I don't know how to show this question. Please help me explaining this question. Thanks a lot. I posted its ...
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83 views

First fundamental form of a surface patch $\sigma$

I have its answer, but I cannot understand again. Please explain its solution clearly. Thanks a lot.
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96 views

A surface patch $\tilde{\sigma}(\tilde u,\tilde v)$ is a reparametrization of a surface patch $\sigma (u,v)$

Suppose that a surface patch $\tilde{\sigma}(\tilde u,\tilde v)$ is a reparametrization of a surface patch $\sigma (u,v)$. Let $$\tilde E d\tilde u^2+ 2\tilde F d\tilde ud\tilde v+\tilde G d\tilde ...
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First fundamental form question.

The question I posted; $6.1.2\quad$ Show that and apply an isometry of $\Bbb R^3$ to a surface does not change its first fund. form. What is the effect of a dilation (i.e., a map $\Bbb R^3\to ...
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The question related to a regular surface.

Prove that an equation of the form $f(x,y,z)=c$ determines a regular surface if $f$, defined on some open subset $S$ of $\mathbb{R}^3$, is smooth and $\nabla f\neq 0$ everywhere in $S$. I know ...
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779 views

what's the difference between isomorphism and homeomorphism?

I think that they are similar or same but am not sure. Can anyone explain the differences?
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71 views

Euler characteristic of part of the sphere

Let R be the part of the sphere in $R^3$ bounded by two smoothly closed curves that do not intersect. For instance, R is the region bounded by a great circle and a smaller circle paralle to it. How ...
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328 views

A non orientable closed surface cannot be embedded into $\mathbb{R}^3$

Can someone please remind me how this goes? Here's the idea of proof I'm trying to recall: let $S$ be a closed surface (connected, compact, without boundary) embedded in $\mathbb{R}^3$. Then one can ...
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63 views

prove that the shapes are isometric

I want to prove that the shapes are isometric. How to prove? There is no info except for the picture. First of all I need to write surface patches. Please can someone help me? The definition of ...
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156 views

prove that $f$ is a diffeomorphism and an isometry

Let $S_1 : [0, 2\pi r]\times [0, h]$ $S_2: x^2+y^2=r^2$ Let $f: S_1 \to S_2$ $(u,v)=(r\cos (\frac{u}{r}), r\sin (\frac{u}{r}), v)$ for $v\in [0,h]$ and $u\in [0, 2\pi r)$ How do I prove that ...
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Differential forms on the torus correspond to periodic forms on $\Bbb{R}^n$?

Let $T^n=\Bbb{R^n/Z^n}$ be the torus. Is it possible say that forms on the torus bijectively correspond to forms on $\Bbb{R}^n$ invariant under translations by integers?
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66 views

Verify this is not orientable.

Verify this is not orientable. Möbius transformation: $$U=\{(t,\theta) \mid \frac{-1}{2}\lt t\lt \frac{1}{2}, 0\lt \theta \lt 2\pi \}$$ $\sigma (t, \theta)=<((1-t\sin (\theta/2))\cos (\theta), ...
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Show that the Mod 2 Linking Number is well defined.

If $X$ and $Y$ are compact, boundaryless smooth manifolds, $f\colon X \to \mathbb R^n$ and $g\colon Y \to \mathbb R^n$ with $f(X) \cap g(Y) = \emptyset$, and with $\dim (X) +\dim (Y) =n-1$. Define ...
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52 views

find f and $d_pf$

Let $$S_1=\{(x,y,z) | x^2+y^2+z^2=1\}-\{N\}$$ $$S_2= \{(x,y,z,0) | x,y\in \Bbb R\}$$ $f:S_1\to S_2$ $f$ is stereographic projection. ,where $\ell$ is a line passing through $N=(0,0,1)$ and $p$ ...
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$d_pf$ is a linear invertible map.

If $f: S_1\to S_2$ is a diffeomorphism, then $$d_pf: T_p(S_1)\to T_q(S_2)$$ is an invertible linear map and $$(d_pf)^{-1}=d_qf^{-1}$$ for any $p\in S_1$ $q\in S_2$ and $f(p)=q$ I cannot prove this ...
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45 views

show that $g\circ f :S_1 \to S_3$ be also smooth and $d_p(g\circ f)=(d_{f(p)}g)\circ (d_pf)$

proposition: Let $S_1$, $S_2$, $S_3$ be 3-regular surfaces and $f:S_1 \to S_2$ and $g: S_2 \to S_3$ be smooth maps. Then show that $g\circ f :S_1 \to S_3$ be also smooth and $$d_p(g\circ ...
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Diffeotopy and connectedness on manifolds

Let $M, N$ manifolds without bundary and $f: M\to N$ and $g: M\to N$ embeddings. We say that a differentiable map $h: [0,1]\times M\to N$ is an isotopy between $f$ and $g$ if each of the maps ...
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$d_pf :T_p(S_1)\to T_{f(p)}(S_2)$ is a linear map for any $p\in S_1$

If $f: S_1\to S_2$ is a smooth map betweentwo regular surface $S_1$ and $S_2$, then $d_pf :T_p(S_1)\to T_{f(p)}(S_2)$ is a linear map for any $p\in S_1$ I cannot prove this statement. I think that ...
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Transversality and intersection mod.2

I am troubled by the following fact: If $f,g : X\to Y$ are homotopic and both transversal to $Z$ then the mod.$2$ intersection numbers are equal $I_2(f,Z)=I_2(g(Z)$. (the book by Guillemin and ...
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What is the type of $u$ in this definition of knot?

I am going to quote from the second paragraph of the Introduction of Möbius Energy of Knots and Unknots by Michael H. Freedman, Zheng-Xu He and Zhenghan Wang. Let $\gamma = \gamma (u)$ be a ...
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Reconstructing a manifold from critical points

I am teaching theoretical calculus this semester, and on the last discussion section we were discussing critical points of functions. I explained the idea of Morse theory, and a student of mine asked ...
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42 views

Standard norm of $\mathbb{R}^3$

I am going through the paper, Energy of a Knot by Jun O'Hara. Let me quote from the Definition 1.1 of Section 1 on the first page: Let $f:S^1 = \mathbb{R}/\mathbb{Z} \to \mathbb{R}^3$ be an embedding ...
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227 views

Smooth diffeomorphisms preserving common symmetries

Let $A$ and $B$ be two $C^\infty$ orientation-preserving diffeomorphic, connected, bounded, open subsets of ${\mathbb R}^n$, with finitely many ends and $G$ the group of isometries of ${\mathbb R}^n$ ...
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1answer
156 views

Prove the directional derivative operators at a point on manifold form a vector space

One of the way to define tangent space is to use directional derivative. However, it's not clear at the first glance that the directional derivative operators form a vector space. Let $D$ be the set ...
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66 views

Constructing a non-degenerate vector-field from old one

This is an attempt to understand Milnor's proof that it's always possible to compute a vector field's index using a non-degenerate field. The strategy is simple: take $z$ to be an isolated zero of the ...
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Existence of differential form on a manifold

I have a fundamental question about the existence of differential forms on manifolds. A $k$-form on a manifold in local coordinates looks like $f(x_1,...,x_n)dx_{i_1}...dx_{i_k}$, where $f$ is a ...
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Why is $\frac{f}{\|f\|}$ a submersion when this matrix has rank $k$?

A paper I'm reading defines the following $\left(\frac{(k-1)k}{2}+1\right)\times n$ matrix: \begin{align*} \Omega_f(x) := \begin{bmatrix} \omega_{1,2}(x)\\ \vdots \\ \omega_{i,j}(x) \\ \vdots \\ ...
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184 views

Definition of a $n$ - form on a manifold

I am confused about the definition of a differential form on a manifold. The definition I have comes from Bott and Tu and is as follows: A differential form, $\omega$, on a manifold $M$ is a ...
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Intersection of lines with compact smooth manifolds

everybody. I need a hint on this: I have to prove that a compact smooth submanifold of R^n intersects almost every one dimensional linear subspace (in R^n) in a finite set of points. I know I have to ...
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The space of regular curves deformation retracts onto the space of arclength parameterized curves

Let X denote the space of smooth maps from the circle into R^3 which have no zero derivatives. This is an open submanifold of the Frechet space of smooth maps from the circle into R^3. Let Y denote ...
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How does a smooth structure on a subset of a manifold determine its status as an immersed submanifold?

As titles are limited to 150 characters, allow me to rephrase my question in a way that is hopefully more precise: Given a $d$-dimensional smooth manifold $M$ and some $k$-dimensional subset ...
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Why is the moduli space of gradient flow lines $\widehat{\mathcal M}(p,q) = \mathcal M(p,q) / \mathbb R$ a smooth manifold?

Let $(M,g)$ be a compact smooth orientable riemannian manifold, and let $f: M \to \mathbb R$ be a Morse function. All functions here are assumed to be smooth. We will be considering the integral ...
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1answer
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If $f\colon X\to S^k$ smooth, $X$ compact and $Z\subseteq S^k$ closed, then $I_2(f,Z)=0$.

I've been working through an old example sheet I found online at Cambridge geometry 2011, to fill in gaps in my topology. The question #14 asks: Suppose $f\colon X\to S^k$ is smooth where $X$ is ...
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70 views

Normal bundle is locally trivial

Could someone tell me how to prove the following result? Let $Z$ be a submanifold of codimension $k$ in $Y$. Prove that the normal bundle $N(Z;Y)=\left\{(z,v):z\in Z, v\in T_{z}(Y)\text{ and } ...
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notation for a torus

I am trying to search for the meaning of this notation but unfortunately it seems that wikipedia even doesn't have it. The book I am following uses the following notation for a torus: $\mathbb{T}^d = ...
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1answer
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Visualizing Frobenius Theorem

Given a smooth vector field $v$ on a (finite dimensional) manifold $M$, one can find the associated integral curves i.e. integral submanifolds of M such that the tangent space at any point $p\in M$ is ...
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Uniqueness of the “asymptotic limit” of a sequence of gradient flow lines

Let $(M,g)$ be a compact smooth orientable riemannian manifold, and let $f: M \to \mathbb R$ be a Morse function. All functions here are assumed to be smooth. We will be considering the integral ...
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105 views

Extending an embedding from a compact submanifold

Suppose $X, Y$ are smooth manifolds, $Z \subset X$ a compact submanifold and $f: X \rightarrow Y$ a smooth map such that the restriction of $f$ to $Z$ is an injective smooth immersion. As $Z$ is ...
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131 views

Intuition of a Submanifold

Could someone explain the intuition behind a submanifold. When, for example, is it appropriate to work with immersed submanifolds vs embedded submanifolds? Why is it important for a submanifold to be ...
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Is there a compact complex manifold with trivial $H_2$?

I don't believe that every complex manifold should have nontrivial $H_2$, otherwise we would easily prove the Chern's conjecture... But the problem is I don't have any counterexample. The Kähler ...
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66 views

Differential topology question involving cobordism

Prove that if $X$ and $Z$ are cobordant in $Y$, then for every compact manifold $C$ in $Y$ with dimension complementary to $X$ and $Z$, $I_2(X, C) = I_2(Z, C)$. [HINT: Let $f$ be the restriction to ...
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3answers
119 views

Showing S^n x R is parallelizable

A manifold $M$ is said to be parallelizable if it admits $k$ linearly independent vector fields. I know that this is equivalent to the tangent space $TM$ being trivial. I am trying to show that ...
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1answer
74 views

Examples of manifolds foliated by $S^2$

I have come across the Frobenius theorem in my study of GR, which for the special case of $S^2$ roughly means, that every point of a manifold with spherical symmetry can be foliated by spheres. I know ...
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Extending a function defined on an arbitrary subset of $\mathbb{R}^n$

This question appeared on an old qualifying exam: Let $X$ be any subset of $\mathbb{R}^n$ and $f\colon X\to\mathbb{R}$ a function with the following property. For every $x\in X$ there is a ...
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345 views

The Poincare Lemma for Compactly Supported Cohomology

I´m reading the proof of The Poincare Lemma for Compactly Supported Cohomology there is a part in the proof that said in the text book Bott and Tu: $d \pi_{\ast} = \pi_{\ast} d$ in other words, ...
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194 views

Are closed, properly embedded manifolds of co-dimension 1 in $\mathbb{R}^n$ orientable?

I have been trying to figure this out as it would seem that it should be so. I have been search though, and the only solution seems to treat the compact case with homology beyond what I know. I ...
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38 views

Degrees are the only interesting intersection numbers on spheres

Show that if $f: X \rightarrow S^k$ is smooth, $X$ compact and $0 < dim(X) < k$, then for all closed $Z \subseteq S^k$ of dimension complementary to $X$, we have $I_2(X, Z) = 0$. An idea:let p ...
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73 views

Hessian quadratic form is well defined

Could someone show why for the Hessian to be well defined ($d_{p}^{2}f(v,w) = L_{v}L_{w}f$) we need $p$ to be a critical point.