Questions tagged [differential-topology]
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.
7,142
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closed but not exact form on $\Bbb{R}^n \setminus \{0\}$.
Let $\omega$ be the $(n-1)-$form on $\Bbb{R}^n \setminus \{0\}$ defined by
$$\omega = \vert \vert x \vert \vert^{-n} \sum_{j=1}^n(-1)^{j-1}x^j \, dx^1 \wedge ... \wedge \, \hat{dx^j} \wedge ... \wedge ...
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Is a monomorphism of vector bundles an embedding?
Let $(p,E,B)$ and $(p',E',B)$ be two $C^r$ vector bundles over the same base space $B$. (When $r>0$ all the spaces are $C^r$ manifolds and all the maps are $C^r$ smooth. When $r=0$ they are just ...
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How do I prove step b in this approach to the prove of the escape lemma
I am trying to prove the the escape lemma (Lee's Intro to smooth manifolds Lemma 9.19)
The proof can be broken down in three parts
a) First prove this lemma:
Lemma: $X$ is a smooth vector field on a ...
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How do I prove the escape lemma?
I am trying to prove the the escape lemma (Lee's Intro to smooth manifolds Lemma 9.19)
The proof can be broken down in three parts
a) First prove this lemma:
Lemma: $X$ is a smooth vector field on a ...
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Surface area of Möbius strip
Since Möbius strips cannot be oriented, the calculation of surface area is meaningless; perhaps that is the reason why I have never seen such problem. However, the following φ is a parameter ...
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Stokes theorem on $C^1$ manifolds?
I have recently encountered Stokes theorem on embedded submanifolds of $\mathbb{R}^n$, and I didn't manage to find a proof for $C^1$ vector fields over $C^1$ manifolds, infact I have only seen that ...
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A lemma to the escape lemma (Lee' Intro to smooth manifolds Lemma 9.19)
I am trying to prove the following lemma, that is used to prove the escape lemma (Lee's Intro to smooth manifolds Lemma 9.19)
Lemma
Suppose
$X$ is a smooth vector field on a smooth manifold $M$ . Let ...
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Closed curve in a torus diffeomorphic to a circle?
In Example 15.9 of Tu's "An Introduction to Manifolds: Second Edition", it is written
Let G be the torus $R^2/Z^2$ and L a line through the origin in R2. The torus can also be represented ...
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Tangent space and derivations on a Banach manifold
When dealing with a finite $d$-dimensional manifold $M$, one can define the tangent space $TM|_p$ of a manifold on a point $p \in M$ in different (but equivalent) ways, based on (at least) the ...
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Tangent map vs. differential on manifolds
Let $E, E'$ be normed vector spaces, $A \subseteq E$ an open set and $f: A \to E'$ a differentiable map. For each point $x \in A$, the differential of $f$ is a linear transformation $Df|_x: E \to E'$. ...
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Is the subset of symmetric $ 3\times 3 $ matrices with given eigenvalues a manifold?
For $ \mu_1>\mu_2>\mu_3 $ are three real numbers. Consider th set
$$
S(\mu_1,\mu_2,\mu_3)=\{A\in M(3,\mathbb{R}):,A^T=A,\,\,\lambda_1(A)=\mu_1,\lambda_2(A)=\mu_2,\lambda_3(A)=\mu_3\},
$$
where $ ...
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Definition of smooth action
Let $G$ be a Lie group and let $M$ be a smooth manifold.
In chapter 7 of Lee's Introduction to Smooth Manifolds, a smooth (left) action of $G$ on $M$ is defined to be a smooth map $\theta:G \times M \...
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How to give an explicit manifold structure on $ \{n\otimes n-m\otimes m:n,m\in S^2,(n,m)=0\} $.
Consider $ N $ defined as follows
$$
N=\{n\otimes n-m\otimes m:n,m\in S^2,\,\,(n,m)=0\}\subset M^{3\times 3},
$$
where $ S^2 $ denotes the two dimensional sphere, $ (\cdot,\cdot) $ represents the ...
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Hopf's proof of the Turning Tangents Theorem in "Differential Geometry in the Large"
Rigorous proofs of the Hopf Umlaufsatz seem to always entail nightmarish details about covering maps and degree theory, or careful estimates as in the discussion here. The strange thing is that Hopf's ...
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Canonical chart for parallelizable manifold
Let $M$ be an $n$ dimensional connected parallelizable manifold and $\{X_1,\dots,X_n\}$ a basis for the tangent bundle. Consider the following
Claim. There existis $p\in M$ such that for all $q\in M$ ...
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A question on the manifold $ \{n\otimes n-m\otimes m:n,m\in S^2,(n,m)=0\} $.
Consider a manifold $ N $ defined as follows
$$
N=\{n\otimes n-m\otimes m:n,m\in S^2,\,\,(n,m)=0\}\subset M^{3\times 3},
$$
where $ S^2 $ denotes the two dimensional sphere, $ (\cdot,\cdot) $ ...
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Differential of a map $\Gamma : M \times N \to P$
I have a smooth map $\Gamma : X\times Y \to Z$ between manifolds with or without boundary, and I am interested in the maps $\Gamma_y : X \to Z$ given by $\Gamma_y(x) = \Gamma(x,y)$. For a given $y\in ...
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Index of a vector field on the unit sphere
Let $F$ be a vector field defined on $\mathbb{S}^2$ such that
$$F(x,y, z)-[ F(x,y, z)\cdot k ]k=F(-x,-y, z)-[F(-x,-y, z)\cdot k]k$$ and $F(x,y,z)=F(-x,-y,-z)$. Here $k=(0,0,1)$.
Indeed $F=(f_1,f_2, ...
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Curvature of the curve on the unit sphere if torsion is always 1. [closed]
Let $I$ be an open interval, $\gamma:I \rightarrow \mathbb{R}^3$ be a curve which is parameterized by arc length and $\gamma(s)$ is in unit sphere for all $s\in I$, and $\tau$ be the torsion of $\...
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$f,g$ are real-valued, $f(x,y)=g(h_1(x),h_2(y))$, $f,h_i$ are continuous, $g$ is increasing, does $g$ must be continuous?
Assumptions: $f,g$ are real-valued. $g:[0,1]^2\to\mathbb R$. Functions $h_1:X\to\mathbb [0,1]$ and $h_2:Y\to\mathbb [0,1]$ are surjective continuous. $X,Y$ are connected separable. $f$ is continuous,
$...
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Relation between the first Pontryagin class $p_1$ and the first Stiefel Whitney class $w_1$ as a tangential structure of manifold M
In Characteristic Class, let us define tangential structure of manifold M such as tangent bundle TM.
Is there a difference between the Stiefel Whitney class $w_1 =0$ and the first Stiefel Whitney ...
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A question related to frame bundle of a vector bundle
I am currently reading up on principal fiber bundles from a set of lecture notes on the subject, and I am trying to make sense of frame bundle of a vector bundle.
Consider a vector bundle $p:E\to M$ ...
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The existence of global continuous frame implies the existence of global smooth frame
I am trying to show that if there exists a global continuous frame for a smooth vector bundle over a smooth manifold $M$, then we can find a global smooth frame.
First, I have proven the simple ...
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Find the points of minimum and maximum curvature
I'm given this 2-D curve: $\gamma(t)=(2cos(t),3sin(t))$ where $t \in [0,2\pi]$. I want to find the points of max and min curvature using calculus, but I keep second guessing my approach... maybe I ...
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Homeomorphisms *as* a subset
Let $X$ be a topological space, let $\mathrm{Map}(X)$ be the set of all continuous maps $X\to X$, equipped with the compact-open topology, and finally, let $\mathrm{Homeo}(X)\subset \mathrm{Map}(X)$ ...
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Smooth map between oriented manifolds.
Let $f: M\rightarrow N$ be a smooth map between smooth closed oriented connected manifolds of same dimension.
Question: is it true that $f$ is smoothly homotopic to some smooth map $g: M\rightarrow N$...
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Defining the Tangent space to the boundary of a manifold $T_p(\partial S)$
While studying manifolds I am having some problem with definition of manifold with boundary.Let $S$ be a regular $n$-level surface in $\mathbb R^{n+1}$ with boundary defined by $S=f^{-1}(0)\cap (\...
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Is the de Rham complex independent of the smooth structure?
The de Rham complex of a smooth manifold $M$ of dimension $n$ is the complex of differential forms $$\cdots\rightarrow 0\rightarrow\Omega^0(M)\rightarrow\Omega^1(M)\rightarrow\cdots\rightarrow\Omega^n(...
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Reference request: Operation on Regular Homotopy Classes
Recently, I stumbled upon the definition of regular homotopy classes and Smale's Theorem as in Smale, S. (1958). Regular curves on Riemannian manifolds. Transactions of the American Mathematical ...
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Degeneration of nodes and punctures on Riemann surfaces
It is well known that it is possible to build Riemann surfaces with singularities such as nodes (like a pinched torus) or with punctures. Now a node on a torus can be obtained by shrinking a circular ...
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Proving $S^{1}$ is a manifold under the quotient topology
Lets say I have defined $S^{1}$ as $[0,1]/\sim$ where $x\sim y$ when $x=y$ except when $x,y$ equal 0 or 1. In other words,
$[0,1]/\sim$ is the singletons and the set $\{0,1\}$. I want to show that ...
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Constructing unique $2:1$ covering of a non-orientable, connected, smooth manifold
Let $M$ be a non-orientable, connected, smooth manifold with $\text{dim }M=n$. I'm trying to fill in the ideas of a construction of a unique $2:1$ covering $\tilde{M}$ of $M$. I've pieced together ...
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Suppose for a region on a surface I can draw a "handle" can I cut the surface to reduce it's genus while leaving the region intact?
Suppose I have a smooth orientable surface $Q$ and a compact region $R$ of $Q$. Suppose there is a closed curve $C$ that divides R into two connected components $R_1,R_2$ but does not divide Q into ...
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Product rule for matrix-vector product
Suppose that $x \in L^2([0,1], \mathbb{R}^m)$ is a vector valued function and $A(x)$ is a ($m \times m$)-matrix whose entries are the components of $x$. Then consider a differentiable curve $\gamma: (-...
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Denseness of $C^{\infty}(M)$ within $C^k(M)$
Let $ M $ be a compact smooth manifold. For $ k \geq 1 $, let $ C^k(M) $ denote the space of real-valued functions on $ M $ of class $ C^k $, equipped with the uniform $ C^k $ norm—that is, the sum of ...
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Every atlas of a smooth manifold has a compatible chart whose image is an open ball
Notation. Given $n\in \Bbb{N}$, $r > 0$ and $x\in \Bbb{R}^n$, let denote $B_r^n(x) = \{ z \in \Bbb{R}^n : d(x,z) < r\}$ where $d$ is the Euclidean metric.
I have to prove the following:
Let $M=(...
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How to determine the fundamental group of $ \{n\otimes n-m\otimes m:n,m\in S^2,(n,m)=0\} $.
Consider a manifold $ N $ defined as follows
$$
N=\{n\otimes n-m\otimes m:n,m\in S^2,\,\,(n,m)=0\}\subset M^{3\times 3},
$$
where $ S^2 $ denotes the two dimensional sphere, $ (\cdot,\cdot) $ ...
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1
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Understunding the inherited atlas
I have to prove the following:
Consider $M:= \{(x,y,z) \in \Bbb{R}^3 : x^2 + y^2 - z^2 = 1 \}$. Prove that $M$ is a smooth manifold and that the map $\Phi : \Bbb{R} \times (0,2\pi) \hookrightarrow M ...
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The approximation of an embedding of a manifold into a Euclidean space with certain conditions
Suppose that $M$ is a compact smooth manifold with dimension $m$, $N$ is a smooth manifold with dimension $n$; $f:M \rightarrow \mathbb{R}^{(m+n+1)}$ is a smooth embedding, $j:N \rightarrow \mathbb{R}^...
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Proving that I have an immersion in a (weak) version Whitney's embedding theorem for compact manifolds
I am proving a weak version of Whitney's embedding theorem for compact manifolds :
Let $M$ be a compact smooth manifold of dimension $m$.
I can be proven that $M$ has a finite smooth atlas $A = \{(...
- 3,025
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Show the level sets are connected
I am trying to prove the level sets of the function
$$
F(x,y) = 2x+2y+\sin(x)+\sin(y)
$$
are connected. I noticed that $\|\nabla F(x,y)\|>0$ for every point of the plane and that $F$ is the sum of ...
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Non-degenerate $2$-form on a manifold gives tangent-cotangent bundle isomorphism
I am trying to solve the following problem:
Let $M$ be a smooth $n$-manifold, and let $\omega\in \Omega^2(M)$ be
such that $\omega_p\colon T_pM\times T_pM\to \Bbb R$ is non-degenerate
for each $p\in ...
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Self-intersection of diagonal vs. alternating sum of Betti numbers.
Let $X$ be a compact, oriented, connected manifold. I recently learned about a cohomological version of Lefschetz' fixed point theorem, which states:
If $f: X \to X$ is a continuous self-map, then $$\...
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Pullback of volume form on $\mathbb{S}^n$ by antipodal map
Let $F:\mathbb{S}^n\rightarrow \mathbb{S}^n$ be the antipodal map, then it's differential $D_pF:T_p\mathbb{S}^n\rightarrow T_{-p}\mathbb{S}^n$ is given by $X\mapsto -X$. Consider the volume form $dx^1\...
- 2,136
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Homoclinic, heteroclinic orbits and nonwandering points
I have a relatively well behaved vector field and I can prove that all the nonwandering points along the flow are equilibria. Is this enough to disprove that there is no homoclinic and heteroclinic ...
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1
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Extending a smooth map from a compact subset to the whole manifold
Let $M$, $N$ be two smooth manifolds (Hausdorff and second-countable). Suppose $K \subset U \subset M$, where $K$ is compact and $U$ is open in $M$. If $f: U \rightarrow N$ is a smooth map, is there a ...
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Homologically trivial embedding of $\mathbb{CP}^2$ into a compact closed smooth $6$-manifold?
Does there exist a compact smooth closed $6$-dimensional manifold $M,$ so that it admits an embedded submanifold $N$ diffeomorphic to $\mathbb{CP}^2$, and $N$ is a boundary in the homology sense? In ...
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Conditions for the derivative of a function to be a proper map
Let $f:U \to \mathbb{R}^{m}$ be a continuously differentiable function, where $U \subset \mathbb{R}^{n}$. In this case, the function
\begin{equation*}
Df:U \to L(\mathbb{R}^{n},\mathbb{R}^{m})
\end{...
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Embeddings and connected sum [closed]
Let $X$ be a compact connected oriented smooth $m$-manifold. For any compact connected oriented smooth $m$-manifold $W$, consider their well-defined connected sum $$X \# W.$$
I wanted to ask two basic ...
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The torus is in $\mathbb{R}^{3}$ or $\mathbb{R}^{4}$?
Always I belived that the torus is in $\mathbb{R}^{3}$, but reading about the clifford Tori I read that the Clifford torus is in $\mathbb{R}^{4}$. I don't understand it. Since in differential ...
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