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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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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234 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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213 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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84 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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153 views

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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220 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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45 views

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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242 views

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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131 views

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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56 views

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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85 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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147 views

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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195 views

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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71 views

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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108 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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156 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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45 views

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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67 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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135 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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77 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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42 views

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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367 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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232 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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1answer
76 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.
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Existence of vector extensions for the Hessian

Q: $p$ a critical point of a smooth $f$ and $v, w$ two vectors in $T_{p}M$. We extend these two to vector fields $v^{*}, w^{*}$ such that at $p$ the first one equals $v$ and the second one equals $w.$ ...
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Volume form on a sphere.

Let $S^n(r)$ be the sphere of radius $r$ , $x_1^2 + ... + x_n^2 = r^2$ and let $$w = \frac{1}{r} \sum_{i=1}^{n+1} (-1)^{i-1} x_i dx_1 \cdots\hat{dx_i},\cdots dx_{n+1} $$ Write $S^n$ for the unit ...
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136 views

Existence of coordinate systems for submanifolds

I decided to do the following problem as an exercise: Let $p \in M$ be a regular point of $f: M \to \mathbb{R}$. Prove the existence of a coordinate system $(x_1,x_2,...,x_n)$ near $p$ such that ...
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[ANSWERED]Lie brackets on vector fields

We consider $v = \frac{\partial}{\partial x}$ and $w = x * \frac{\partial}{\partial z} + \frac{\partial}{\partial y}$. I need to first find the Lie bracket between them which i get to be: ...
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levels curves of polynomial equations as manifolds

Q: For which real values c is the subset $f(x) = x_{1}^{2} + x_{1}^{3} - x_{2}^{2} + x_{3}x_{4} = c$ a smooth submanifold of $\mathbb{R}^4$? Try: for it to be a smooth submanifold, $c$ has to be a ...
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320 views

Continuous function approximation on manifolds

I am asked to show that every cont. function from a manifold M to $\mathbb{R}$ can be approximated by smooth functions. Try: let f be a map from M to the reals(R). Let ${s_{i}, U_{i}}$ be our atlas. ...
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Whitney Embedding theorem for manifold with boundary in Lee's Introduction to smooth manifolds(2nd Edition)

I am reading through the new edition of Lee's book and I am stuck by the proof of Theorem 6.15. When passing to a non-compact manifold, the author begin by defining several sub-levelsets and claim ...
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how to obtain a generalized Morse function out of a fiber bundle?

Let $M\to E\to B$ be a smooth fiber bundle. In "Parametrized Morse Theory and Its Applications,(Proceedings of the ICM, 1990)", K. Igusa says that if dim $B$$<$dim $M$, then, there exists a smooth ...
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a condition for smooth vector field

Let $M$ be a Hausdorff manifold. I'm trying to prove that a vector field $Y:M\to TM$ is smooth if and only if the derivation induced by $Y$ for all globablly defined smooth functions is smooth. That ...
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A Milnor Differential Topology Excercise

If $m<p$, show that every map $f:M^m\longrightarrow\ S^p$ is homotopic to a constant, where $M^m$ is smooth manifold of dimension $m$. I tried to show that $M^m$ is contractible or convex, but I ...
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174 views

Why are normal bundles always locally trivial?

Is there a quick and dirty proof that normal bundles (say of some submanifold in a smooth manifold) are always locally trivial? My notes seem to have swept this assumption under the rug. Even ...
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Moving a compact submanifold off of another submanifold?

This is an intuitive idea that I see referenced a lot. Consider the following situation. Let $M$ and $N$ be submanifolds, $M$ compact, in some larger manifold $X$. Suppose also that ...
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78 views

Baire Category Theorem in a Smooth Manifold

Let $Z\subset M$ be a set of measure-0 , in a [smooth] manifold $M$. How does one shows that $M$ \ $Z$ is everywhere dense in $M$, using Baire category theorem? and which of the theorem version is ...
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52 views

Continuity of point wise orientation

Prove that $[(X_1,...,X_n)]$ on a manifold $M$ is continuous if and only if every point $p$ in $M$ has a coordinate neighborhood $(U,\phi) = (U,x^1,...,x^n)$ such that for all $ q \in U$, the ...
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Analytic subvariety in complex manifold

I am trying to figure out a statement in a textbook "If $M$ is any complex manifold of a projective space $\mathbb{P}^{n}$, $V\subseteq M$ an analytic subvariety of dimension $k$, then we can find a ...
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$h$-principle for isometric embeddings

All the references I have seen so far list the Nash $C^1$-embedding theorem as an example where the $h$-principle holds. The $h$-principle for a differential relation holds by definition, when the ...
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157 views

Trivialisation of Möbius strip

I've just started studying Advanced Geometry and I'm in trouble with a (stupid) exercise. It's about finding a trivialisation of the Möbius strip (I'll refer to it as $E $) viewed as a fibre bundle ...
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89 views

Constructing vector bundles from local covers and transitions functions

Let $M$ be a smooth manifold. Suppose we are given an open cover ${U_\alpha}$ of $M$ and for $\alpha,\beta$ ; a smooth map $\tau_{\alpha\beta}\colon U_\alpha \cap U_\beta \to GL(k; R)$ satisfying the ...
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82 views

The sheaf of smooth functions is soft

A sheaf $\mathcal{F}$ over some topological space $X$ is called soft if any section over any closed subset can be extended to a global section (i.e. over $X$). How to understand that the sheaf of ...
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216 views

How does a left group action on the fiber of a principal bundle induce a right action on the total space?

Suppose I define a "principal $G$-bundle" as follows: A principal $G$-bundle is a fiber bundle $F \to P \overset{\pi}{\to} X$ with a left group action of $G$ on $F$ that is free and transitive, ...
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Proof that a set $X \subset M$ is a Manifold

Let M be a manifold without boundary and let , $g:M\to \mathbb R$ have $0$ as a regular value. Than the set $X \subset M$ with $g(x) \ge 0$ is a smooth manifold with boundary equal to $g^{-1}(0)$. I ...
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86 views

Smooth function with equibounded family of derivatives

By $\mathcal{C}^{\infty}(\mathbb{R})$ we denote the space of smooth functions $\mathbb{R}\rightarrow \mathbb{R}$. Also, by $\mathrm{supp}(f)$ we denote the closure $\mathrm{Cl}(f^{-1}(\mathbb{R}_{\ne ...
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Why do Zoll metrics exist only on $S^2$ and $RP^2$?

Zoll metric on a Riemannian manifold is a metric for which all geodesics are closed and have the same period. For sure, a standart metric on the sphere $S^2$ has this property: all its geodesics are ...