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.

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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 ...
anonymous's user avatar
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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 ...
darkside's user avatar
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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 ...
Blue Various's user avatar
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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 ...
Lorenzo Vanni's user avatar
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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 ...
darkside's user avatar
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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 ...
Foo's user avatar
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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 ...
Pedro G. Mattos's user avatar
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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 $Df|_x: E \to E'$. ...
Pedro G. Mattos's user avatar
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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 $ ...
Luis Yanka Annalisc's user avatar
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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 \...
Joseph Kwong's user avatar
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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 ...
Luis Yanka Annalisc's user avatar
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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 ...
Hopf-Appreciator's user avatar
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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 $q\in M$ ...
Gabriel Golfetti's user avatar
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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) $ ...
Luis Yanka Annalisc's user avatar
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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 ...
Nick F's user avatar
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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, ...
Matchmaticians's user avatar
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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 ...
zeta's user avatar
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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 $p:E\to M$ ...
neophyte's user avatar
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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 ...
ImmUyd90210's user avatar
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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 $\mathrm{Homeo}(X)\subset \mathrm{Map}(X)$ ...
Jan Bohr's user avatar
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11 votes
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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 $g: M\rightarrow N$...
GSM's user avatar
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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 (\...
Kishalay Sarkar's user avatar
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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(...
Martin Frenzel's user avatar
1 vote
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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 ...
daniele de gennaro's user avatar
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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 ...
Fra's user avatar
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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 ...
Manseej Khatri's user avatar
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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 ...
Rough_Manifolds's user avatar
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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 ...
Hao S's user avatar
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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: (-...
motionart's user avatar
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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=(...
Superdivinidad's user avatar
2 votes
1 answer
254 views

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) $ ...
Luis Yanka Annalisc's user avatar
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1 answer
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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 ...
Superdivinidad's user avatar
2 votes
1 answer
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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}^...
Quzs's user avatar
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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 = \{(...
some_math_guy's user avatar
2 votes
1 answer
72 views

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 ...
Dadeslam's user avatar
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1 vote
1 answer
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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 ...
Random's user avatar
  • 587
1 vote
0 answers
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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 $$\...
red_trumpet's user avatar
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1 vote
0 answers
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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\...
Chris's user avatar
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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 ...
giangian's user avatar
  • 257
2 votes
1 answer
63 views

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 ...
Skyskie's user avatar
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16 votes
1 answer
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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 ...
Zhenhua Liu's user avatar
3 votes
0 answers
44 views

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{...
ProphetX's user avatar
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0 answers
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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 ...
Terry Black's user avatar
1 vote
0 answers
77 views

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 ...
Skinner.'s user avatar
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