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In mathematics, a parallelization of a manifold $M$ of dimension n is a set of n global linearly independent vector fields.

Formal definition

Given a manifold $M$ of dimension n, a parallelization of $M$ is a set $\{X_1,\dots ,X_n\}$ of n vector fields defined on all of $M$ such that for every $p\in M$ the set $\{X_1(p),\dots ,X_n(p)\}$ is a basis of $T_{p}M$, where $T_{p}M$ denotes the fiber over $p$ of the tangent vector bundle $TM$.

A manifold is called parallelizable whenever admits a parallelization.

Examples

• Every Lie group is a parallelizable manifold.
• The product of parallelizable manifolds is parallelizable.

Properties

Proposition. A manifold $M$ is parallelizable iff there is a diffeomorphism $\varphi :TM⟶M\times {\mathbb{ℝ}}^n$ such that the first projection of $\varphi$ is ${\tau }_M:TM⟶M$ and for each $p\in M$ the second factor—restricted to $T_{p}M$—is a linear map ${\varphi }_p:T_{p}M\to {\mathbb{ℝ}}^n$.

In other words, $M$ is parallelizable if and only if ${\tau }_M:TM⟶M$ is a trivial bundle. For example suppose that $M$ is an open subset of ${\mathbb{ℝ}}^n$, i.e., an open submanifold of ${\mathbb{ℝ}}^n$. Then $TM$ is equal to $M\times {\mathbb{ℝ}}^n$, and $M$ is clearly parallelizable.

See also

• Chart (topology)
• Differentiable manifold
• Frame bundle
• Orthonormal frame bundle
• Principal bundle
• Connection (mathematics)
• G-structure
• Web (differential geometry)

Notes

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References

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• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
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