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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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 ...
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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 $v = tx$ ...
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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 ...
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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 $\mathbb R^n$. ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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: X = A^a(x,v)\frac{\partial}{\partial x^a} + B^a(x,...
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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 $(x, y)$ ...
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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 ...
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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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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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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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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\...