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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57 views

Show that there is a $\pi_i$-related smooth vector field for each smooth vector field $X_i \in \Gamma(M_i,TM_i)$

Assume $M_1, \dots,M_k$ are smooth manifolds and define $M:=M_1\times \dots \times M_k$. Denote the projections on the $i$-th factor with $\pi_i: M \rightarrow M_i$. I want to show that for each ...
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Why do we use tangent bundles to define vector fields on manifolds?

Most textbooks define a vector field on a smooth manifold $M$ as a section of the tangent bundle of $M$. My question is: why is it even necessary to talk about bundles when defining vector fields on $...
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Differential using sheaves

I am trying to understand smooth real manifolds using sheaves and finding some trouble with differential of smooth maps. Some notation: Let $M$ be manifold, I denote by $\mathcal{O}_M$ it's sheaf of ...
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What are differential forms?

For a manifold $M$, if we want to speak of "tangent vectors," we often say the tangent bundle $TM$ is the space of tangent vectors. This is sort of an abuse of terminology, I guess you could say, ...
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Calculating the tangent bundle of $O(m)$ and $SL(m)$.

I want to show that (a) $TO(m) = \{(M,MA) \mid M \in O(m), A \in T_{Id}O(m)\}$ and (b) $TSL(m) = \{(M,MA) \mid M \in SL(m), A \in T_{Id}SL(m)\}.$ By definition, $TX = \{(x,y) \mid x \in X, y \...
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Does every smooth local frame of the tangent bundle correspond to a chart?

Every smooth chart $(U,\phi)$ on a smooth manifold $M$ determines a smooth local frame $U \to TM$ on the tangent bundle, namely $(\partial/\partial x^i)$, where $(x_i)$ are the coordinate functions of ...
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Charts and atlas for a tangent bundle (stereographic projection)

I’m reading Walter Thirring’s treatment of tangent spaces in Classical Mathematical Physics and have been struggling with the meaning of some of the abstract notation. Specifically, I don’t understand ...
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A question about the global representation of a tangent vector field.

Here is the question. Assume that $X$ is a smooth tangent vector field of $M$, and $X(p)=0$. Show that there exists finitely several smooth functions $f_i$ and smooth tangent vector fields $X_i$, ...
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What does the sentence “chart induces coordinates” really mean?

Let $Q$ be a smooth $n$-manifold and let $(U,\varphi)$ a chart for $Q$. What does "$\varphi$ induces local coordinates $(q^1,...,q^n)$? I supposed $\varphi=(q^1,...,q^n)$, i.e. $$\forall x\in Q, \quad ...
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Differential of $(x,v)\mapsto (x,\exp_x(v))$

Let $M$ be a pseudo-Riemannian manifold. Let $\Omega=\{(x,v)\in TM: |v|<\epsilon\}$. I want to compute the differential of the map \begin{align*} \Omega &\to TM\\ (x,v) &\mapsto (x,\exp_xv)...
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74 views

The map $f:T_1M\to \hat{\mathbb{R}}$ is continuous.

Let $(M,d)$ be a complete Riemannian manifold and let $T_1M=\{v\in TM: \|v\|=1\}$. Define a map, $$s:T_1M\to \hat{\mathbb{R}},~~ s(v)= \sup\{t:d(\pi(v),\operatorname{exp}(tv))=t\}$$ where $\pi$ is the ...
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Does the existence of the cross product relate to the triviality of $TS^n$?

The dimensions $n$ for which there exists a cross product in $\mathbb{R}^n$ are exactly those for which the tangent bundle $TS^n$ of the $n$-sphere is trivial (i.e. $TS^n = S^n \times \mathbb{R}^n$): $...
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Projectivization of the tangent bundle to $S^2$ is nontrivial

I want to prove that the projectivization of the tangent bundle to $S^2$ is nontrivial. It looks very similar to the hairy ball theorem. Also, I would like to understand if the projectivization of ...
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Infinitesimal symmetry of a tangent distribution.

An infinitesimal symmetry of a tangent distribution $D={\rm span}(X_1,\dots,X_n)$ on $n$-dimensional manifold $M$ is a vector field $Y$ such that for every $X\in D$ a Lie bracket $[X,Y]\in D$. My ...
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Tangent bundle: disjoint union

In textbooks about differential geometry, one finds often the disjoint union in the definition of the tangent bundle (e.g. in "Lee: Intorduction to smooth manifolds", or "Amann, Escher: Analysis III"):...
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Find the Jacobian determinant of $\tilde \psi \circ \tilde \phi ^{-1}$ at $\phi (p)$

This is the exercise 12.2 of An Introduction to Manifolds of Loring Tu: Let $(U,\phi )=(U,x^1,\ldots ,x^n)$ and $(V,\psi)=(V,y^1,\ldots ,y^n)$ overlapping charts on a manifold $M$. Then they induce ...
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Equivalence between derivation of $C^\infty$ algebra and derivative?

A dervation of $C^\infty$ is a linear operator $D:C^\infty\mapsto C^\infty$ s.t. $$D(fg)=(Df)g+f(Dg).$$ It is clear that the derivative operator $\frac{d}{dx}$ is a derivation. But (up to what extend) ...
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Tangent and cotangent bundle as an associated bundle

In a book I read the isomorphisms below were mentioned without any explanation. Is there any intuitive way to see these identities hold? Let $M$ be a smooth $n$-manifold, $F(M)$ a frame bundle of $...
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Is there a global definition of the tangent bundle?

The tangent bundle of a smooth manifold is usually defined by equipping the disjoint union of the tangent spaces with a smooth structure. Is there a way to define the tangent bundle as a vector bundle ...
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Intuition of normal bundle to a manifold

So i recently learned about fibre bundles and tangent bundles in particular. While the definition of tangent bundles seems quite intuitive, i really struggle to understand any definition of the ...
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Why defining tangent bundle?

I'm learning a bit about smooth manifolds, and currently I'm learning about tangent bundles (just definitions mainly) and vector field. This is my reference : Tu's Introduction to Manifolds. I was ...
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Fiber of the Hitchin fibration

In the paper "More On Gauge Theory And Geometric Langlands" by Edward Witten ( https://arxiv.org/abs/1506.04293 ), he writes about the Hitchin fibration $\pi: \mathcal{M}_H \rightarrow \mathcal{V}, (...
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Tangent bundle for smooth algebraic variety

I was discussing with some friend about how to define the tangent bundle for a smooth variety since this is very natural in the manifold setting and we couldn't find references discussing in detail. ...
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Let $D$ be a distribution. Show that every point of $p \in M$ has a submanifold $N$ such that $p \in M$ and $T_p N = D_p$.

Let $D$ be a distribution. Show that every point of $p \in M$ has an embedded submanifold $N$ such that $p \in M$ and $T_p N = D_p$. Attempt: Assume $M$ is a smooth manifold of dimension $m$ and $D$ ...
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An exercise in differential topology

Problem: Given a smooth submanifold $M\subset\mathbb{R}^k$, show that the tangent bundle space $$TM=\{(x,v)\in M\times\mathbb{R}^k:v\in TM_x\}$$ is also a smooth manifold. Show that any smooth map $f:...
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$\mathbb{P}^2(\mathbb{R})$ can't be a smooth fiber bundle of a $1$-dimensional smooth manifold

This question was stated during a lesson where I asked my professor if every two dimensional smooth manifold arises as a tangent bundle of some one dimensional smooth manifold. My professor said that ...
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Lagrangian submanifolds, symplectomorphism

I want to show the following: Let $L_0,L_1 \subset \mathbb{R}^{2n}$ be Lagrangian submanifolds (standard symplectic structure on $\mathbb{R}^{2n}$. Let $p=(x,y) \in \mathbb{R}^{2n}$, s.t. $p \in L_0 \...
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What is the fiber of the tangent bundle?

A fiber bundle is a space $(E,B,\pi,F)$ such that $\pi:E\rightarrow B$ and $E$ locally looks like the product space $B\times F$. If M is a smooth manifold, the tangent bundle on M is the space $(T(M),...
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Are vector fields really sections of the tangent bundle?

Let $(TM,M,\pi,F)$ be the tangent bundle on $M$. I've seen multiple resources state that a vector field on $M$ is a section in $\Gamma_{TM}(M)$ of the tangent bundle. A vector field is a map $$\Phi:M\...
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50 views

Corresponding flow on cotangent bundle

I have an exercise where it states: Let $Y:Q \rightarrow TQ$ be a complete vector field with flow $\varphi_t$. Let $X : T^{*}Q → T(T^{*}Q)$ be the vector field generating the corresponding flow $\...
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Existence of dense velocity curve on connected manifold

Let $M$ be a non-empty, connected manifold. Show that there exists a differentiable curve $\gamma : \mathbb{R} \to M $ , so that the image of the velocity curve $\dot{\gamma} :\mathbb{R} \to TM$ is ...
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Explanation for the relation of a metric tensor in a vector space and the metric function on topological spaces

I have always wondered, an it seems as I cannot figure it out on my own, the following. Consider a differentiable manifold $M$ with a metric tensor $g$, which acts on tangent vectors at any point $p\...
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Clarifying some details about Orientability of Surfaces using Vector Fields

For an orientable surface embedded in $\mathbb{R}^3$, we can properly define a normal vector field on it, and we can't do so on a nonorientable surface. On the other hand, there is a result saying ...
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Degree of canonical bundle

I'm reading theorem IV.5.3 in Hartshorne book and he says : let $f: X \longrightarrow X^{\prime}$ be a canonical morphism of hypereliptic curve. Suppose $\mu=deg(f)\geq2$ and $d=deg(X^{\prime})$ than ...
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Connection 1-forms?

I'm working through Lee's "Riemannian Manifolds: an Introduction to Curvature" by myself, and I'm a bit stuck on problem 4-5: Let $\nabla$ be a connection on $M$, let $\{E_i\}$ be a local frame on ...
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For a differentiable retraction $f$, there is a local coordinate system in which $f$ is the canonical projection.

Let $M$ be a n-dimensional manifold and $f: M \to M$ be a differentiable retraction, that is $f\circ f=f $. Let $p\in f(M)$. Show that there is a chart $(\phi, U)$ of $M$, around $p$, such that $\phi \...
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If $M$ be a submanifold of $N$, is true that $TM$ is a submanifold of $TN$? [duplicate]

Today, the professor of Smooth Manifold's course proofed that $TS^{2n-1}$ is an embedded submanifold of $T\mathbb R^{2n}$ using the fact that $T\mathbb R^{2n}$ is diffeomorphic to $\mathbb R^{2n} \...
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Integrable distribution on a Lie group coming from a homogeneous space

Let $M=G/H$ be a homogeneous space of a Lie group $G$ and a closed subgroup $H$. Consider the principal bundle $\pi:G\to G/H=M$. Let $\mathcal D\subset \Gamma(TM,M)$ be an integrable tangent subbundle ...
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Transition functions of the tangent bundle of a projective variety.

If we work (for example) in the category of differentiable manifolds, then i saw that it is standard calculating the transition functions of the tangent bundle of a differentiable manifold. It seems ...
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Question about diffeomorphisms of tungent bundles.

I'm studying the chapter one of this text. In the end of the chapter there are a few exercises. The exercises 32 to 35 asks to constructs natural diffeomorphisms that maps fibers to fibers. The ...
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lifting vector field on $M$ to its tangent bundle $TM$

Is there a canonical way of lifting vector field $X$ on $M$ to its tangent bundle $TM$? I came up with this question while studying tangent bundle formalism of Lagrangian mechanics. In Lagrangian ...
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Trying to Find Intuition Behind Why Riemannian Metrics are Defined on The Tangent Space of Manifolds

I'll preface this by noting that I'm looking for a more intuitive answer rather than a formal argument here. I just started taking Riemannian Geometry, and I'm a little hung up on why Riemannian ...
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176 views

Tangent Space for Torus

I'm trying to find the basis for $T_{p}M$ for the torus $T^{2}= S^{1}$x $S^{1}$ so, my atlas for the manifold is \begin{equation} \varphi(u,v) = ((r\cos{u}+a)\cos{v},(r\cos{u}+a)\sin{v},r\sin{u}) \...
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Differential structures of sum bundle, tangent bundle, dual bundle, tensor bundle, quotient bundle, etc

As the title suggests, I was wondering what structure a certain vector bundle $E$ should induce in its "derived" bundles such as the sum bundle, the dual bundle, the tensor bundle, the tangent/...
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Symplectic reduction and the isotropy group

In Berndt's $\textit{Introduction to Symplectic Geometry}$ we have the following statements: Assume we are given a symplectic manifold $(M, \omega)$, a symplectic operation $\phi:G\times M \...
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Pullback of connection on tangent bundles. [duplicate]

Let $f:M\rightarrow N$ be a smooth map; surjective submersion and all that is necessary to call it a well behaved map. Let $\Gamma$ be a connection on $N$ (a connection on the tangent bundle $TN\...
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How to put the manifold structure on $\wedge^2 M$?

Let $M$ be a smooth manifold with an atlas $\{(U,\varphi)\}$, and let $\wedge^2 M$ be the disjoint union of $A^2(T_p M)$, where $T_pM$ is the tangent bundle of $M$ at $p$ and $A^2(T_p M)$ is the ...
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Is a vector field a section or a tangent vector?

Is a vector field a section or a tangent vector? This question arose when I read two statements in a book named geometric control of mechanical system. The first ...
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second order vector field on TQ

I have 2 nomeclature questions: Does a vector field on tangent bundle, $V\in \mathfrak{X}(TQ)$ is the same as second order vector field? Or the second order vector field are those that arise from ...
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Quotient of tangent bundle

Let $M\to B$ be a principal $G$-bundle. Then I have a group action of $G$ on $M$. I think, that this induces a group action of $G$ on the tangent bundle $TM$ of $M$ by looking at the differentials of ...

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