Questions tagged [tangent-bundle]
The tangent $TX$ of a smooth (real or complex) manifold is defined as disjoint union of all the tangent space at the points of $X$. This the first and natural example of vector bundle.
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vector bundle $\pi : E \rightarrow M$ and any $x \in M$: $\forall v \in E_x \exists s \in \Gamma(E) $s.t $s(x) = v$
I am trying to prove this
For any vector bundle $\pi : E \rightarrow M$ and any $x \in M$:
$\forall v \in E_x \exists s \in \Gamma(E)$ s.t $s(x) = v$
$\Gamma(E)$ is the space of (smooth) sections of ...
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Topology on the tangent space TM: Is this really the initial topology?
In a lecture series, I have come across the statement that the topology on the tangent space $TM$ is given by the coarsest topology which makes the projection map $\pi: TM \mapsto M$
continuous.
In ...
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Spivak' proof that möbius band has not trivial tangent bundle.
I'm reading Spivak's Comprehensive introduction to differential geometry and i came across the proof that the tangent bundle of the Möbius band (as he defines it at an early stage i presume) is not ...
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Inducing a Lie algebra action from a Lie group action
Let $G$ be a Lie group, $M$ be a (smooth) manifold and assume that $G$ acts smoothly on $M$ with a fixed point $p \in M$. (I mean, there is a Lie group homomorphism $\rho : G \to C^{\infty}(M,M)$ ...
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Why is the abstract functorial definition of the tangent bundle not widely accepted?
The following quote from page 595 of Spivak's Calculus exemplifies my viewpoint on definitions:
It is an important part of a mathematical education to follow a construction of the real numbers in ...
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Prove that a vector field $X$ is smooth if and only if its coordinates (or components) $X_i$ are smooth for all chart of manifold $M$
Question: Prove that a vector field $X$ is smooth if and only if its coordinates (or components) $X_i$ are smooth for all chart of manifold $M$.
Solution: Assume that $X$ is smooth. Take any chart $(U,...
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What properties define the tangent bundle up to diffeomorphism?
In the theory of smooth manifolds there are many ways in which the tangent bundle can be defined, begging the question: what set of properties define the tangent bundle 'up to diffeomorphism'?
These ...
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Let $M$ be a smooth manifold. Does there exist a canonical isomorphism between $\Gamma(TM)^*$ and $\Gamma(T^*M)$?
Let $M$ be a smooth manifold and for each point $p \in M$, let $T_pM$ denote the tangent space at $p \in M$. We define the set $TM = \bigsqcup_{p \in M}T_pM$ and equip it with the initial topology and ...
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The tangent bundle is a smooth manifold
I am reading the proof that the tangent bundle can be made into a smooth $2n$-manifold from Introduction to Smooth Manifolds by John M. Lee, and wanted to ask for clarification on something.
I'm on ...
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Why Representation of Clifford algebra are constant for an orthonormal frame?
Let $e_\alpha$ be a basis of the tangent bundle $TM$ and $ \rho: T_x M \rightarrow \operatorname{End}\left( W\right)$ a representation of a Clifford algebra.
In this text Field theory from a bundle ...
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When does the shape operator commutes with a differential?
Suppose we have a smooth map $\varphi : S\to H$ between two regular parametric surfaces in $\mathbb{R}^3.$ Then at any point $p\in S,$ we have following maps between corresponding tangent spaces:
$\...
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On Marsden's 'Introduction to Mechanics and Symmetry' Exercise 4.3-3. (orientability of product manifold)
Given two differentiable manifolds $\mathcal M$ and $\mathcal N$ I needed to show that
$$\mathcal M, \; \mathcal N \mbox{ orientable } \Rightarrow \mathcal M \times \mathcal N \mbox{ orientable.}$$
...
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Is the isomorphism type of $TM$ as a topological vector bundle determined by the homeomorphism type of $M$? [duplicate]
Let $M, N$ be smooth manifolds with the same underlying topological manifold $X$. Are $TM$ and $TN$ isomorphic, regarded as topological vector bundles over $X$?
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Derivative of a curve on tangent bundle $TQ$ and second-order equations (where is the acceleration?)
This is very basic question that I should have resolved long ago but didn't and it still plagues me.
Let $TQ$ be the tangent bundle of some (configuration) manifold, $Q$, and let $(\pmb{q},\pmb{v}) = (...
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Nonequivalent fiber bundles with the same total space, base space, and fiber
This answer gives an example of two fiber bundles with the same total space and base space but topologically distinct fibers. Is there an example where the fibers are homeomorphic too, but the bundle ...
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direction tangent bundle of $S^2$ homeomorphism
Let $\triangle S^{2}$ ( the tangent direction bundle of $S^2$) be the unit tangent bungle of $S^2$ where the opposite tangent directions are identified.
Let G be the (order four) group of quaternions ...
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Marked points on fibers of $TS^2$ can trace out a helix during (parallel) transport.
It is claimed here that a mark on a rod parallelly transported by an observer moving along a geodesic $\gamma(t)$ on a smooth manifold $M$ (with a metric $g$) will trace out a helix, instead of a ...
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On Bishop's 'Tensor Analysis on Manifolds' Problem 6.3.3 (canonical lift into double tangent bundle ends up in secondary bundle structure)
I have a two questions regarding this exercise:
Are there any mistakes in my solution attempt?
If one speaks of a vector field on the double tangent bundle, but does not specify which bundle ...
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Normal space of orthogonal matrix.
Let $ O(n) $ be the manifold of orthornormal matrix, i.e.
$$
O(n)=\{A\in\mathbb{R}^{n\times n}:A^TA=I\}.
$$
Then $ O(n) $ is a submanifold of $ \mathbb{R}^{n\times n} $. On $ O(n) $, there is a ...
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Coordinate transformations of the tangent bundle as a manifold.
My question can be summarized as: Let $\mathcal{M}$ be a smooth manifold and $T\mathcal{M}$ be its tangent bundle. It's well known that $T\mathcal{M}$ can be viewed as a smooth manifold. Then how does ...
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Tangent bundle of a fibered product
There is an argument that I would like to fully understand, but I can't see it, yet.
Here is the situation: Given smooth manifolds $X$, $Y$ and $Z$ and transverse maps $f:X\rightarrow Z$, $g\...
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Tangent Space under (linear) Transformation
i am looking for a confirmation of the following Lemma as well as a reference:
Let $M \subset \mathbb{R}^m $ be a smooth-manifold and $A \in \mathbb{R}^{n\times m}$ be a full rank matrix with $n\geq m$...
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Obstructions to smooth extending the smooth distribution from the boundary.
1. Suppose $(M, \partial M)$ be an $n$-dimensional manifold with boundary, and suppose $E$ be a $(k-1)$-dimensional subbundle of $\mathrm{T} (\partial M)$.
Then, could we always find a $k$-dimensional ...
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Does there exist $X$, $k$, $M$ such that $N(X)$ is diffeomorphic to open punctured unit ball.
For a $k$-manifold $X$ in $\mathbb{R}^M$, define its tangent bundle $T(X) \to X$ and the normal bundle $N(X) \to X$. Let $B$ denote the open punctured unit ball in $\mathbb{R}^3$, i.e., $B = \{y \in \...
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Pullback of tangent bundle of codomain induced by a submersion is a quotient of tangent bundle of domain
Let $M,N$ be smooth manifolds and $f: M \to N$ a smooth map. Then, we have the following commutative diagram
Here, $\pi_1, \pi_2$ are the bundle projections from the tangent bundles, $p_1, p_2$ are ...
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The tangent bundle as an embedded submanifold when $M$ is an embedded manifold
If $M$ in an $m$-dimensional embedded submanifold of $\mathbf{R}^2$, then $TM$ is an embedded sub manifold of $T\mathbf{R}^n$ because we have the normal bundle $NM$ is an embedded sub manifold and $TM$...
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How are transition functions of $TM $ defined when $ M$ is an oriented manifold?
This question was asked in my quiz of Differential geometry course and I am having a really hard time in this course.
Question: Let $M$ be an oriented manifold of dimension $n$ and let $π : TM \to M$ ...
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Equivalence of homotopical definition of spin structures with the "classical" one
Let $M$ be an oriented smooth surface, and let \begin{equation}\mathbb{Z_2} \to \operatorname{Spin}(2) \overset{\theta}{\to} \operatorname{SO}(2) \end{equation} be the usual central extension defining ...
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Criterion for existence of vector fields spanning $T_pM$ for all $p\in M$
Given a diffeomorphism $F:M\times\mathbb{R}^n\longrightarrow TM$ such that $F(\{p\}\times\mathbb{R}^n)=T_pM$ for all $p\in M$ and for every $p\in M$ the map $v\in\mathbb{R}^n\mapsto F(p,v)\in T_pM$ is ...
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Equivalent definitions of 1-forms
Let $M$ be a manifold, then a 1-form $\omega$ is a function that assigns to each point $p \in M$ a linear function or covector. This definition seems natural to me as it exactly captures a 1-form ...
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Natural boundary chart of the tangent bundle
in his "Introduction to Smooth Manifolds", 2.ed., p.67, John M. Lee poses the following exercise, after he has outlined how a natural chart of the tangent bundle TM is constructed from an ...
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Definitions of Flat Connections
I have some questions about the definitions of flat connections, and want to make sure if my understanding is correct.
In Jost's Riemannian Geometry and Geometric Analysis, two definitions of flatness ...
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Trivial tangent bundle not diffeomorphic to 2n-dimensional open set
I was trying to find an explicit example of a trivial tangent bundle, i.e. $TM = M\times \mathbb{R}^n$, with $M$ a smooth manifold without boundary of dimension $n\in\mathbb{N}$, which is not ...
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A (maybe) trivial question on trivial vector bundles: alternative definition of trivial vector bundle
I am studying Loring W. Tu's Differential geometry, Connections, Curvature and Characteristic Classes and I am having a doubt (the same doubt I had when studying the same topic in the author's An ...
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Torsion on a flat connection (Geometric Intuition).
Start with the 2-sphere $\mathcal{S}^2$ with the standard $(\theta, \phi)$ chart ($\theta = 0$ is the North pole etc) and metric:
$$ds^2 = d\theta^2 + \sin^2 \theta d\phi^2$$
Remove the two poles from ...
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Natural coordinates on tangent bundle
In Lee's Introduction to Smooth Manifolds, he says that
...the tangent bundle of a smooth manifold has a natural structure as a smooth manifold in its own right. The natural coordinates we ...
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What is the universal property of the thickening $Y[\varepsilon]$?
Given an $S$-scheme $Y$, let $Y[\varepsilon]$ denote thickening $Y[\varepsilon]=Y \times_S D_S$ of $Y$. Here $D_S$ is the $S$-scheme $D\times_\mathbb Z S\to S$ where $D = \operatorname{Spec} \mathbb{Z}...
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Motivating the standard cotangent bundle Lie group structure
If $G$ is a Lie group with product $\circ: G \times G \to G$, an "obvious" Lie group structure present on the tangent bundle $T G$ is given by taking the differential of the Lie group ...
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Homology groups of Klein bottle's unit tangent bundle.
Let $K$ denote Klein bottle and $T^1K$ its unit tangent bundle. I want to compute homology group of $T^1K$, I've seen this discussion: Homology groups of unit tangent bundle, I don't understand much ...
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Why are the tangent spaces $T_pM$ and $T_qM$ disjoint for $p \neq q$?
Let $p, q \in M$, where $M$ is some smooth manifold. Then, according to Tu's book Introduction to Manifolds the tangent spaces $T_pM$ and $T_qM$ are disjoint. Of course we define the tangent bundle $...
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Holomorphic tangent bundle over $\mathbb{C}P^1$
Denote the universal bundle of $\mathbb{C}P^1$ by $U$, the dual bundle by $H$. Prove that the Holomorphic tangent bundle over $\mathbb{C}P^1$ is isomorphic to $H\otimes H$, and calculate the dimension ...
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Differential of a smooth map in terms of sheaves of deriviations
$\newcommand{\C}{\mathscr{C}^\infty}$$\newcommand{\blank}{{-}}$$\newcommand{\from}{\colon}$$\newcommand{\after}{\circ}$$\newcommand{\Der}{\mathrm{Der}}$Recall that for a smooth manifold $M$, there is ...
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Do tensors in the tangent space act on functions and vectors?
I know via isomorphism we may treat the tangent space of a point on a manifold as the vector space of derivations on functions at that point. I.e. we can give the tangent space the basis of partials:
$...
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Are tangent spaces at different points disjoint according to this definition?
In this lecture by Prof. F. P. Schuller (I've included the correct time stamp) it is claimed that two tangent spaces at different points are disjoint, i.e. $T_pM\cap T_qM =\emptyset$ for $p,q\in M$ ...
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Complex manifolds that do not admit holomorphic foliations by curves with discrete singular sets
A singular holomorphic foliation by curves of a complex manifold $M$ is generated by a nonzero global section $\sigma$ of $TM \otimes L$, for some line bundle $L \to M$. If we choose $L$ correctly, ...
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Why is the frame bundle of $S^1$ homeomorphic to $S^1$?
I'm trying to understand why there are only 2 spin structures on the circle $S^1$. From my understanding, in dimension 1 at least, a spin structure on a manifold $M$ is just a double cover of the ...
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For Hirzebruch Surfaces, does the tangent exact sequence split?
Consider the projection $\pi:\mathbb{F}_n \rightarrow \mathbb{P}^1$. Is it true that the following exact sequence $$0 \rightarrow T_{\pi} \rightarrow T_{\mathbb{F}_n} \rightarrow \pi^*T_{\mathbb{P}^1} ...
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What type of quantity is absement?
In differential geometry, position is a point in a manifold, velocity is a vector in the tangent bundle, and acceleration is a quantity in the double tangent bundle (or the tangent bundle if a ...
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How is acceleration connected to the second order tangent bundle?
The top answer to What does the integral of position with respect to time mean? says that “Acceleration is an element of 𝑇𝑀 (if we use a connection to identify the horizontal subbundle of 𝑇(𝑇𝑀) ...
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Does a twisted tangent bundle $T\mathbb P^n \otimes \mathcal O(d-1)$ ever have a globally nonvanishing section?
Let $X = \mathbb P^n$ be the usual projective space over an algebraically closed field. For what values of $d \in \mathbb Z$ does the twisted tangent bundle $E = TX \otimes \mathcal O_X(d-1)$ have a ...
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