# 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 ...
1 vote
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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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### 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 $$\mathcal M, \; \mathcal N \mbox{ orientable } \Rightarrow \mathcal M \times \mathcal N \mbox{ orientable.}$$ ...
1 vote
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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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### 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 $n\geq m$...
1 vote
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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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### 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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### 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 ...
### 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 ...