information:filtering
Manifolds with boundary

If M is a manifold with boundary, then an orientation of M is defined to be an orientation of its interior. Such an orientation induces an orientation of ∂ M. Indeed, suppose that an orientation of M is fixed. Let U → R be a chart at a boundary point of M which, when restricted to the interior of M, is in the chosen oriented atlas. The restriction of this chart to ∂ M is a chart of ∂ M. Such charts form an oriented atlas for ∂ M.

When M is smooth, at each point p of ∂ M, the restriction of the tangent bundle of M to ∂ M is isomorphic to T ∂ M ⊕ R, where the factor of R is described by the inward pointing normal vector. The orientation of T ∂ M is defined by the condition that a basis of T ∂ M is positively oriented if and only if it, when combined with the inward pointing normal vector, defines a positively oriented basis of T M.