Definition:Orientable Manifold
 | This article needs to be tidied. In particular: usual Please fix formatting and $\LaTeX$ errors and inconsistencies. It may also need to be brought up to our standard house style. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Tidy}} from the code. |
 | This article needs to be linked to other articles. In particular: usual You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding these links. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{MissingLinks}} from the code. |
 | This page has been identified as a candidate for refactoring of basic complexity. In particular: usual Until this has been finished, please leave {{Refactor}} in the code.
New contributors: Refactoring is a task which is expected to be undertaken by experienced editors only. Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$.To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Refactor}} from the code. |
For a finite-dimensional vector space
If $V$ is a finite-dimensional vector space, an orientation of $V$ is an equivalence class of ordered bases for $V$,
where two ordered bases are considered equivalent if:
- the transition matrix that expresses one basis in terms of the other has positive determinant.
.gif)
Every vector space has exactly two orientations.
Once an orientation is chosen, a basis is said to be positively oriented if it belongs to the chosen orientation, and negatively oriented if not.
The standard orientation of $\R^n$ is the one determined by the standard basis $\left(e_1, \ldots, e_n\right)$, where $e_i$ is the vector $(0, \ldots, 1, \ldots, 0)$ with 1 in the $i$ th place and zeros elsewhere.
For a smooth manifold
If $M$ is a smooth manifold, an orientation for $M$ is a choice of orientation for each tangent space that is continuous in the sense that:
- in a neighborhood of every point there is a (continuous) local frame that determines the given orientation at each point of the neighborhood.
If there exists an orientation for $M$, we say that $M$ is orientable.
 | Work In Progress You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by completing it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{WIP}} from the code. |
Also see
- Results about smooth manifolds can be found here.
Sources
- 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.): Appendix B Review of Tensors: Differential Forms and Integration