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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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Confusion on the vector fields and tangent bundles

Definition of vector fields and tangent vectors I use here: Let $M$ be a smooth manifold. Given $p\in M$, here the tangent vector $X\in T_pM$ is defined to be a linear map from$C^{\infty}(M)\to \...
Bowei Tang's user avatar
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Tangentbundle of submanifold of $\mathbb{R}^n$

Let $M\subseteq \mathbb{R}^n$ be a $k$-dim submanifold of $\mathbb{R}^n$, and I want to prove that the tangent bundle of $M$ is a submanifold of $\mathbb{R}^n$. The idea is: Since $M$ is a submanifold ...
Gao Minghao's user avatar
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Hirsch tangent bundle definition

I am new to differential topology, and I attempt to study it using Hirsch book "Differential Topology". In the first chapter, after introducing the definition of manifold, he talks about ...
Anna  Vakarova's user avatar
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1 answer
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An analogue of the de Rham complex for vector fields

Let $M$ be a smooth manifold and let $\chi$ be its tangent bundle and $\Omega^1$ be its bundle of $1$-forms. Using the exterior algebra we can extend the space of sections of $\Omega^1$ to a ...
Zoltan Fleishman's user avatar
3 votes
1 answer
81 views

Structure of a homogeneous vector bundle on the tangent bundle T(G/B)

Let $G$ be a reductive algebraic group over the field $\mathbb C$, $B\subset G$ a Borel subgroup and $G/B$ their quotient variety. For every $B$-module $(M,\rho)$ we can construct the associated $G$-...
Vereinsmeister's user avatar
1 vote
0 answers
29 views

Unable to visualize geodesic as an integral curve of a vector field in the tangent bundle.

I have began to study Riemannian Geometry and there I encountered a statement that I am unable to feel/understand/visualize. It is the following statement that is still bothering me: Any geodesic for ...
Kishalay Sarkar's user avatar
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59 views

Cartesian product of two manifolds and tangent spaces

I have an $n$ dimensional parallelizable manifold $M$. I know that it is the cartesian product of two parallelizable manifolds $M_1$ and $M_2$ but I do not know these two manifolds, not even their ...
Doriano Brogioli's user avatar
2 votes
2 answers
270 views

Tangent bundle of a sphere $T\mathbb S^n$ is diffeomorphic to $\mathbb S^n \times \mathbb S^n - \Delta$

Let $\mathbb S^n$ denote the $n$-sphere, which is the smooth manifold consisting of all points in $\mathbb R^{n+1}$ with Euclidean norm one. Recall that the tangent bundle of $\mathbb S^n$, denoted $T ...
Joseph Kwong's user avatar
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2 answers
70 views

Intrinsic proof that sprays induce involutions

Let $M$ be a smooth manifold. Let $V$ be the canonical vector field on $T M$ (also called the Liouville vector field), which if $(x, y)$ are local coordinates on $T M$ is defined by $V = y^i \frac{\...
Keeley Hoek's user avatar
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Affine connection induced by connection map $K: TTM \to TM$

I'm currently reading Riemannian Geometry by Saski, and just finished reading Proposition 4.1 regarding the connection map of a an affine connection. In particular, suppose that $M$ is a smooth ...
infinitylord's user avatar
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Computing transition functions from trivial Whitney sum

Suppose $E_1$ and $E_2$ are two vector bundles over a base $B$ such that $E_1 \oplus E_2 \cong \epsilon$, i.e., the sum is trivial. After taking intersections, we get a cover $U_i$ of $B$ such that $...
JZweifler's user avatar
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2 votes
1 answer
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Decomposition of the tangent bundle of a tensor product of vector bundles

If $E \to B$ is a smooth vector bundle, then the tangent bundle $T E$ is a vector bundle over both $E$ and $T B$. (If you like, the latter structure is the derivative of the vector bundle structure on ...
Keeley Hoek's user avatar
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Tangent space of $RP^n$ vs the orthogonal complement to the line bundle $\gamma_n^1$

I am reading Milnor's lectures on characteristic classes. He defines the canonical line bundle $\gamma_n^1$ as the set of points $(\pm x, v) \in \mathbb{R}P^n \times \mathbb R^{n+1}$, where $v = tx$ ...
JZweifler's user avatar
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Isomorphism between Tangent Sheaf and Cotangent Sheaf

I'm studying K3 surfaces and I often encountered the fact that the tangent sheaf $\mathcal{T}$ is isomorphic to the sheaf of differentials $\Omega$. Why is this true? I guess this follows from the ...
WindUpBird's user avatar
3 votes
1 answer
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Questions on notation that I haven't seen before (tensor product sign in superscript and mysterious $\Gamma$ symbol)

I have some questions about the Wikipedia article on Tensor fields. The definition of a tensor field is given as such: Let $\mathfrak M$ be a manifold, for instance the Euclidean plane $\mathbb R^n$. ...
Nate's user avatar
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2 votes
2 answers
118 views

What is a transversal intersection? Can it be explained without tangent spaces and tangent bundles?

What is a transversal intersection? Can it be explained without tangent spaces and tangent bundles? Background: Transversal intersection was used to explain If the interior of two convex manifolds ...
SRobertJames's user avatar
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79 views

Confused about horizontal and vertical lift of 1-forms

Let $(Q,\pmb{g})$ be an $n$-dim Riemannian manifold with Levi-Civita connection/covariant derivative $\nabla$. I believe I understand the vertical and horizontal lifts a vector field on $Q$ to a ...
J Peterson's user avatar
1 vote
1 answer
90 views

Hairy Ball Theorem on $TS^1$ and $TS^2$

Edit: It seems my language was not correct. By "orientable", I mean there are no non-zero sections of the tangent bundle, i.e. the sphere fails the Hairy Ball Theorem, while the circle does ...
Fly by Night's user avatar
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3 votes
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What exactly is $T^2f$?

Given a smooth map $f : M \to N$ between manifolds, the differential gives a map $Tf : TM \to TN$ between tangent bundles. Taking another differential gives a map $T^2f : T^2M \to T^2N$ between ...
Frank's user avatar
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A problem about the tangent bundle of manifold and submanifold

If $M$ is a regular submanifold of $N$,then for any $x \in M$, use local coordinate,we can naturally take $T_xM$ as a subspace of $T_xN$, now I want to prove $TM$ is a submanifold of $TN$, where $TM,...
ckx's user avatar
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3 votes
1 answer
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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
-4 votes
1 answer
220 views

Why am I wrong about the tangent bundle? [duplicate]

What is gained by insisting on the distinction between tangent spaces at different points and double-tagging them in the construction of the tangent bundle? What specifically am I missing by ...
R. Burton's user avatar
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1 vote
1 answer
85 views

Relation of connections on $TM$ and $T^*M$

I have troubles following the book "Elements of Noncommutative Geometry". Let $E\to M$ be a vector bundle. Then a connection on $E$ is a linear map $\nabla:\Gamma^\infty(M,E)\to\Gamma^\infty(...
Schrödinger's cat's user avatar
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79 views

Uniqueness of Levi-Civita connection in terms of horizontal distributions

Lately, I have been wondering about how to make the reasoning of uniqueness of the Levi-Civita connection $\nabla$ on a tangent bundle $\pi:TM\to M$ with a bundle metric $g$, via f.e. the Koszul ...
whatever's user avatar
1 vote
0 answers
48 views

Is it easier to construct global sections on tangent bundle then other fiber bundle?

When talking about differential forms, it seemst that it is not a serious problem about whether can there be forms on a manifold $M$ at all - it's like they are just functions from some domain and you ...
jw_'s user avatar
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0 answers
92 views

Push forward of a vector field on the tangent bundle TM onto M under the bundle projection map

Given the tangent bundle $\pi:TM \rightarrow M$, a generic vector field $X \in T_{0}^{1}(TM)$ can be written in local coordinates as: \begin{equation} X = A^a(x,v)\frac{\partial}{\partial x^a} + B^a(x,...
Roberto Ricci's user avatar
1 vote
1 answer
60 views

Pulling back tangent vectors to basis the tangent bundle

Suppose we have a differentiable manifold and coordinate chart $M \xrightarrow{\phi=(x, y)} \mathbb R^2$ Let $f: M \rightarrow \mathbb R$ be a $C^\infty M$ function specified "in $(x, y)$ ...
Mark's user avatar
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2 votes
0 answers
56 views

Why fiber bundles are locally products

I'm a first year graduate student in math and I'm currently studying fiber bundles. The definition is clear and I understand how it generalize concepts as (co)tangent bundles or vector bundles. What ...
Gesture Glove's user avatar
1 vote
1 answer
138 views

Explaining the tangent bundles of $S^n$ for $n=1,3,7$

I have seen several posts on the tangent bundles of $S^1$, and $S^3$. Basically, it seems that the idea is to find $n$ linearly independent smooth vector fields. In particular, for $S^1$, we pick $x\...
Ook's user avatar
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71 views

Frankel exercise on coordinate invariant vector fields on $TM$

I am doing the following exercise from Frankel's The geometry of physics and I am having some doubts: 2.3(2) Consider the tangent bundle to a manifold $M$. (i) Show that under a change of coordinates ...
l4teLearner's user avatar
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64 views

A flawed proof that tangent bundles are diffeomorphic to the product of the manifold and $\mathbb{R}^n$.

I was asked for a homework question to prove that $\mathbb{S}^1 \times \mathbb{R}$ is diffemorphic to $T\mathbb{S}^1$. Intuitively, it was clear to me how this could be the case, but when i attempted ...
Soze's user avatar
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0 answers
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Reference on identifying the differential of a map with a $(1,1)$ tensor field

I've seen several times on the Internet the pretty cool expression that \begin{equation}\label{1}\mathrm df=\sum_{i,j}{\partial f^i\over \partial x^j}{\partial \over \partial y^i}\otimes\mathrm dx^j\...
painday's user avatar
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0 votes
1 answer
58 views

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 ...
some_math_guy's user avatar
1 vote
1 answer
95 views

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 ...
P.Jo's user avatar
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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 ...
No Signals's user avatar
4 votes
1 answer
172 views

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 $\rho : G \to C^{\infty}(M,M)$ ...
Luka's user avatar
  • 126
12 votes
2 answers
731 views

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 ...
Baylee V's user avatar
  • 594
1 vote
1 answer
154 views

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,...
N00BMaster's user avatar
1 vote
1 answer
107 views

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 ...
Sam's user avatar
  • 5,166
1 vote
1 answer
130 views

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 ...
Druizr's user avatar
  • 127
1 vote
0 answers
159 views

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 ...
Muhannad Al Ayoubi's user avatar
0 votes
0 answers
32 views

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 ...
amilton moreira's user avatar
2 votes
0 answers
94 views

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: $\...
Bumblebee's user avatar
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1 vote
0 answers
21 views

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 $X$?
isekaijin's user avatar
  • 1,755
2 votes
2 answers
199 views

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}) = (...
J Peterson's user avatar
5 votes
0 answers
84 views

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 ...
Karl's user avatar
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1 vote
1 answer
81 views

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 ...
Pellenthor's user avatar
1 vote
0 answers
99 views

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 ...
Alfons Winkel's user avatar
0 votes
2 answers
106 views

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 ...
Luis Yanka Annalisc's user avatar
3 votes
1 answer
113 views

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\...
Teddyboer's user avatar
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