Tagged Questions

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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20 views

Submersion and immersion

I googled wiki about submersion and immersion. Wiki states that submersion is dual to immersion. I wonder where this duality relationship comes from.
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32 views

The intuition of the rank theorem

In Rudin's Principle of Math Analysis, I know the proof of the rank theorem. However, I fail to understand its content. 9.32 Theorem: Suppose $m,n,$ are nonnegative integers, $m\geq r,n\geq r$, $F$ ...
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1answer
27 views

Finding bump function on a smooth manifold using partitions of unity.

Let $M$ be a smooth manifold. Let $A$ and $B$ be disjoint closed sets of $M$. Show there exists a smooth function $f$ such that $f^{-1}(0)=A$ and $f^{-1}(1)=B$. This is my idea so far, Since $A$ ...
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26 views

Exponential map on Diffeomorphism group of $S^1$

I am reading Segal book on Loop groups, and he mentions the following theorem: $$ \exp: Vect(S ^1) \rightarrow Diff(S ^1) $$ the map taking a vector field to the diffeomorphism obtained by flowing ...
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17 views

Retract of a free $\Omega(\mathbb{R})$-module

Can an open subset X of $\Omega(\mathbb{R}^2)$ be an $\Omega(\mathbb{R})$-module retract of some free $\Omega(\mathbb{R})$-module? Here $\Omega(\mathbb{R}^n)$ denotes the usual topology of ...
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1answer
75 views

A question about contractibility of a Lie group

In Lawson's book Spin Geometry, chapter II, before Proposition 1.11, it mentions a fact that if the 1st, 2nd and 3rd homotopy groups of a Lie group $G$ vanish (although $\pi_2(G)=0$ automatically), ...
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1answer
91 views

Prove that $f$ map a neighborhood of $Z$ diffeomorphically on to a neighborhood of $f(Z)$

Supposed that the derivative of $f:X\to Y$ is an isomorphism whenever $x$ lies in the sub-manifold $Z \subset X$, and assume that $f$ maps $Z$ diffeomorphically onto a $f(Z)$ . Prove that $f$ map a ...
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97 views

De Rham Cohomology is Sheaf Cohomology

I'm trying to prove that de Rham cohomology computes the cohomology of the constant sheaf $\mathbb{R}$ of real valued smooth functions on a manifold. I would like to do this without using Čech ...
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2answers
28 views

Prove that if $S^k$ has a non vanishing vector field, then its antipodal map is homotopy to the identity if $k$ is even.

Prove that if $S^k$ has a non vanishing vector field, then its antipodal map is homotopy to the identity if $k$ is even. Here is what I got so far. Suppose we have an antipodal map $x \to -x$ of ...
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2answers
41 views

Prove this plane algeraic curve is not a differentiable manifold

Prove the algebraic curve $\{(x,y)~|~x^2(x+1)-y^2=0\}$ in $\mathbb{R}^2$ is not a differentiable manifold. Remark: It is evident that the given cubic curve has a singularity at $(0,0)$ which disable ...
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1answer
47 views

Show that if $f_0$ is Morse in some neighborhood of a compact set $K$, then so is every $f_t$ for $t$ is sufficiently small.

Suppose that $f_t$ is a homotopic family of function on $R^k$. Show that if $f_0$ is Morse in some neighborhood of a compact set $K$, then so is every $f_t$ for $t$ is sufficiently small. I know that ...
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1answer
39 views

Prove that $f$ is Morse function if an only if $det(H)^2 + \sum_{i=1}^k (\frac {\partial f}{\partial x_i})^2>0$

Let $f$ be a smooth function on an open set $U\subset R^k$. For each $x \in U$ let $H(x)$ be the Hessian Matrix of $f$, whether $x$ is critical point or not. Prove that $f$ is Morse function if an ...
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1answer
48 views

Existence of smooth diffeomorphism $f$ of open ball onto itself with $f(0) = p$.

I am trying to show that for every point $p$ of the open $n$-disk $B^n$, there exists a smooth diffeomorphism $B^n \to B^n$ sending $0$ to $p$. Certainly, it seems intuitively obvious for points $p$ ...
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1answer
58 views

Naturality of the pullback connection

I'm completely stuck proving the naturality of the pullback connection. The strategy suggested is a follows: We let $\phi: (M,g) \to (\tilde{M}, \tilde{g})$ be an isometry, with connections ...
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1answer
47 views

Noncomplete riemannian manifolds

In Lee's Riemannian manifolds text, he claims that "on $\mathbb{R}^n$ with metric $(\sigma^{-1})^* g$ obtained from the sphere from stereographic projection, there are geodesics that escape to ...
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1answer
38 views

Divergence in terms of Levi-Civita connection

The divergence of a vector field $X$ on a manifold $M$ is defined usually as the function $\text{Div}(.)$ such that $(\text{Div} X) \;\mu =L_X \mu$ for $\mu$ a volume form. I know that there is also ...
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1answer
41 views

The derivative as a linear function

In Milnor's Topology From the Differentiable Viewpoint, the derivative of a smooth map $f: U\to V$ is defined as $$ \mathrm{d}f_x: \mathbb{R}^k \to \mathbb{R}^l $$ $$ h\mapsto \lim_{t\to0} ...
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0answers
33 views

Diffeomorphism between $\mathbb{R}^{2}/\sim$ (Torus ) and $\mathbb{S}_{1}\times \mathbb{S}_{1}$

This is homework so no answers please. Consider the quotient map $\pi:\mathbb{R}^{2}\to \mathbb{R}^{2}/\sim$ ,where $(x_{1}y_{1})\sim (x_{2},y_{2})$ iff $x_{1}-x_{2},y_{1}-y_{2}\in \mathbb{Z}$. I ...
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13 views

Frobenius theorem for singular foliations , what are hypotheses impose over an endomorphism $P:TM\to TM$ that it span a singular foliation in M?

Let $M$ a smooth manifold. Given a morphism $P:TM\to TM$, i.e, $\pi\circ P\equiv Id$ and is linear over the fibers. If we suppose that $P^2=P$, then for each $x\in M$, $P_{x}:T_xM\to T_xM$ is a ...
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22 views

Tangent bundle to 2-sphere isnt trivial as a vector bundle?

I read many quiestion about $TS^{2}\ncong S^{2}\times\mathbb{R}^{2}$ where the hint is use Hairy ball theorem and directly is done. My question is: how do I proof that $TS^{2}\ncong ...
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1answer
34 views

Calculate Brieskorn Manifold?

I need show that Brieskorn Manifold is submanifold with dimension $2n-1$ and calculate specifically for $d=2$ and $n=1$ $W(d)=\lbrace (z_{0},z_{1},...,z_{n})\in \mathbb{C}^{n+1}\vert$ $ ...
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1answer
106 views

A question regarding the proof of Hopf's theorem.

This is a question regarding the proof of the Hopf theorem given in "Topology from a Differential Viewpoint" by Milnor: If $v:X\to \Bbb{R}^m$ is a smooth vector field with isolated zeroes, and if ...
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1answer
38 views

Equivalence between orientation of the tangent bundle and orientation of manifolds

If $M^{n}$ is a manifold then the following statement are equivalent. The tangent bundle $(TM,\pi,M)$ is an orientable $n$-dimensional vector bundle. $M$ has an $\lbrace (U,h)\rbrace$ atlas on $M$ ...
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28 views

Showing the quotient map $\pi:\mathbb{R}^{2}\to \mathbb{R}^{2}/\sim$, where $x_{1}-x_{2},y_{1}-y_{2}\in \mathbb{Z}$, is open

This is homework so no answers please. Showing the quotient map $\pi:\mathbb{R}^{2}\to \mathbb{R}^{2}/\sim$, where $(x_{1}y_{1})\sim (x_{2},y_{2})$ iff $x_{1}-x_{2},y_{1}-y_{2}\in \mathbb{Z}$ is ...
2
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1answer
66 views

Intuitive idea of immersions?

So I understand the definition of immersions and submersions, as well as the motivation for defining such ideas. Not only are they important in understanding properties of mappings of tangent spaces, ...
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0answers
36 views

why is there no non-degenerate 2-forms on 4-sphere?

The question is in the title. I have been told that there are actually no non-degenerate 2-forms on $S^{2n}$ for $n \neq 1,3$. I have found the following question: No symplectic structure on ...
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1answer
30 views

A question from Milnor's “Topology from a differentiable viewpoint”

Milnor's "Topology from a Differentiable Viewpoint" says the following: Let $f:M\to N$ be a smooth mapping, where $M$ is $m$ dimensional and $N$ is $n$ dimensional. Moreover, $m\geq n$. If $y\in ...
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1answer
13 views

The differential of a smooth map on manifold at points of local maxima

I have a differentiable function $f:M \to \mathbb{R}$ where $M$ is a smooth manifold. If $p \in M $ is a point of local maxima, that is I have an open set $V \subset M$, $p \in V$, so that $f(p)\geq ...
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1answer
17 views

Invariant forms on principal bundles

Let $\pi:M \to B$ be a principal $G$-bundle and $\xi$ a invarint $k$-form on $M$. Does $k> dimG$ implies that $\xi$ is a basic form (pull back of a $k$-form on the base manifold $B$)?
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1answer
43 views

Is this a valid way to show $\chi(SL_n(\mathbb{R}))=0$?

Why does $\chi(SL_n(R))=0$? I'm going about it like this. Let $X:=SL_n(R)$. Define a map $f:X\to X$ such by $A\mapsto BA$, where $B$ is the identity matrix, except with an extra $1$ in the upper ...
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0answers
31 views

Given points $x,y$, and tangent vectors $v,w$, is there a diffeomorphism $x\mapsto y$ and $v\mapsto w$?

Suppose $M$ is a smooth manifold, and you're given two points $x,y$, and associated tangent vectors $v$ and $w$. I'm curious, does there exist a diffeomorphism $F$ such that $F(x)=y$ and $dF_x(v)=w$? ...
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1answer
54 views

Representative of a cohomology class in once punctured solid torus

Consider a once punctured solid torus $(\mathbb R^2 \times S^1) /\{pt\}$. It is not difficult to see that it is homotopy equivalent to the bouquet of spheres $S^2\vee S^1$. So this guy has a ...
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0answers
40 views

Naturality of Lie derivatives

It was left to me as an exercise to show the naturality of the Lie derivative of arbitrary tensors on a smooth manifold. Is the following argument correct (it seems too easy)? Let $X$ be a vector ...
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1answer
43 views

Orientable surface bundles over the circle and their structure group

I want to understand whether orientable surface bundles over the circle, i.e. with orientable total space, are always trivial, so I though I would revive an old post and ask for a few clarifications, ...
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1answer
47 views

Degree of a vector field on the boundary of a punctured ball?

Suppose $v\colon \mathbb{R}^3\setminus\{x_1,\dots,x_n\}\to S^2$ is a smooth map, where all the points $x_i$ sit inside the unit ball $B^3$, and not on the boundary $S^2$. I suppose this map can be ...
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1answer
68 views

rudin's principles of mathematical analysis 10.31

I'm working on rudin's principles of mathematical analysis(3rd edition). There is problem too complex for me to solve.Please help me. The problem is on p270-271 of text. "Let T be 1-1 mapping of ...
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1answer
45 views

A doubt from Milnor's “Topology from a Differentiable Viewpoint”.

This is a doubt from Milnor's "Topology from a Differentiable Viewpoint". For a smooth $f:M\to N$, with $M$ compact, and a regular value $y\in N$, we define $n(f^{-1}(y))$ to be the number of ...
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1answer
38 views

Is this orientation preserving or reversing?

I am confused about the definition of orientation on manifolds. Let $X=\{(x,y,0)\in \mathbb{R}^3 \mid x^2+y^2=1\}$ and $Y=\{(x,y,1)\in \mathbb{R}^3 \mid x^2+y^2=1\}$ be two one dimensional circles in ...
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1answer
26 views

Trivial nature of orientable fibre bundle with cylinder base space

Given a bundle whose base space is a "cylinder", i.e. $\mathbb{S}^1 \times \mathbb{B}^n$ where $\mathbb{B}^n$ is an $n$-ball, is orientability (of the total space) enough to ensure that the bundle is ...
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1answer
34 views

Differential Structure of Two Dimensional Manifold

I am a beginner in Differential Topology, and have learnt that Theorem One dimensional differential manifold without boundary is diffeomorphic to $\mathbb R$ or $\mathbb S$. So I am curious ...
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1answer
89 views

Is there anything “nice” about the set of normal matrices (over $\Bbb R$ and $\Bbb C$?)

Normal matrices are of course useful to any linear algebra buff, not least because of the spectral theorem. However, taken as a whole, they tend to have some not-so-nice properties. For example: ...
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0answers
27 views

Orientability of space with only even dimensional cells.

I have the following question: Suppose a compact $\mathbb{R}$-manifold has finite cell decomposition with only even dimensional cells. Then $M$ is orientable. It's a theorem that a closed ...
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0answers
43 views

Transitivity of smooth submanifolds

I was reading through Guillemin and Pollack and was having trouble verifying this for myself. Given $M \subset N$ and $N \subset P$, where $M$ is a submanifold of $N$, and $N$ a submanifold of $P$, ...
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1answer
37 views

Simultaneous coordinate representation of a submanifold and its sub-submanifold

Suppose $Z\subset X\subset Y$ are manifolds and $z \in Z$. Prove that there exist an independent function $g_1,...g_l$ on a neighborhood $W$ of $z$ in $Y$ such that $$Z \cap W =\{y\in ...
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1answer
48 views

Prob. 10 of Chapter 8 of Milnor's Topology From a Differentiable Viewpoint. (Tangent Bundle is a Smooth Manifold.)

I want to solve Prob. 10 of Chapter 8 from Milnor's Topology From a Differentiable Viewpoint. The problem states that: Let $M\subseteq \mathbf R^k$ be a smooth manifold of dimension $m$. Show ...
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1answer
42 views

Does the cup product on de Rham cohomology induce a nondegenerate bilinear form?

I have small issue I came across in the following. Suppose $M$ is a compact, oriented manifold of dimension $4n+2$. I want to prove that the de Rham cohomology group $H^{2n+1}(M)$ are even ...
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1answer
56 views

Pullback of a form under the retraction $r\colon \mathbb{R}^n\setminus\{0\}\to S^{n-1}$.

The following is from Spivak's DG Lemma 7 in Chapter 8, but I'm muddled in a computation. Define two $(n-1)$-forms on $\mathbb{R}^n\setminus\{0\}$ by $$ ...
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1answer
30 views

How can we prove that $GL(2,\mathbb{R})$ is a topological group in $R^{4}$

A group is called topological group if it satisfies three properties 1) G is a Hausdorff space in K (Here we want to prove that if $ A,B \in GL(2,\mathbb{R})$ then we can find two disjoint open sets ...
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1answer
24 views

inverse of stiefel-whitney class of product bundles

Let $\xi_1,\cdots,\xi_n$ be vector bundles over $M$. Let $\omega$ be the Stiefel-Whitney class and $\bar\omega$ be the inverse. By an exercise in Milnor's book, $$ w(\Pi_{j=1}^a ...
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0answers
23 views

Orientation of the intersection of manifolds

From Guillemin and Pollack Differential Topology: Compute the orientation of $\mathbf{X}\cap\mathbf{Z}$ in the following examples by exhibiting positively oriented bases at every point: a) ...