Knowlpedia

Definition

Boundary orientation

The orientation on a manifold boundary determined by placing an outward normal before a positive boundary basis.

Definition

Let MM be an nn-dimensional . Its ∂M\partial M has the boundary orientation defined as follows. At p∈∂Mp\in\partial M, an ordered basis (v1,…,vn−1)(v_1,\ldots,v_{n-1}) of Tp∂MT_p\partial M is positive exactly when

(ν,v1,…,vn−1)(\nu,v_1,\ldots,v_{n-1})

is a positive basis of TpMT_pM for any outward-pointing vector ν∈TpM\nu\in T_pM. Replacing ν\nu by another outward-pointing vector does not change the sign, so the rule is well defined. This is the outward-normal-first convention.

Role in Stokes' theorem

With this convention, has no additional sign:

∫Mdω=∫∂Mω\int_M d\omega=\int_{\partial M}\omega

for every compactly supported (n−1)(n-1)-form ω\omega. The orientation is therefore not decorative data: reversing it changes the boundary integral and breaks this formula Lee, Chapter 16.

Basic examples

Give the interval [a,b][a,b] its orientation from the increasing coordinate. Its oriented boundary is {b}−{a}\{b\}-\{a\}: the positive orientation at bb is +1+1, while at aa it is −1-1. For an oriented product [a,b]×N[a,b]\times N, the two boundary faces inherit opposite orientations, with the precise product sign determined by the order of the interval and NN factors.

Conventions and scope

Some texts use inward-normal-last or outward-normal-last conventions. These are equivalent only after the appropriate dimension-dependent sign is inserted. A boundary orientation requires an orientation of MM; an orientable but unoriented manifold does not determine one canonically.

References
  1. John M. Lee, Introduction to Smooth Manifolds, 2nd ed., Springer, 2012. DOI record. Relevant: Chapter 16, boundary orientations and Stokes' theorem.
  2. Loring W. Tu, An Introduction to Manifolds, 2nd ed., Springer, 2011. DOI record. Relevant: Chapter 23, orientation of the boundary.