# 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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 ...
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 ...