Definition:Oriented Manifold

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Definition

An oriented manifold is a smooth orientable manifold together with a choice of orientation.

If $M$ is an oriented $n$-manifold, then a smooth coordinate chart $\left(U,\left(x^i\right)\right)$ is said to be an oriented chart if the coordinate frame $\left(\partial / \partial x^1, \ldots, \partial / \partial x^n\right)$ is positively oriented at each point.

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Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.): Appendix B Review of Tensors: Differential Forms and Integration