Inward, outward, and boundary-tangent vectors

Depends on

• Smooth invariance of the manifold boundary
• Tangent and cotangent bundles extend over a boundary

Used by

• An interval has nonzero Euler characteristic despite being odd-dimensional Counterexample
• An inward field without a negative-time flow in the half-line Counterexample
• An inward radial field violates the outward boundary formula Counterexample
• Induced boundary orientation Definition
• Morse function adapted to a cobordism Definition
• The closed half-space as a manifold with boundary Example
• The outward radial field on a disk Example
• The boundary stable tangent bundle splits off a trivial line Lemma
• The index sum of an outward field is the Gauss degree Lemma
• The index sum of an outward field on an even-dimensional manifold Lemma
• The reflection of an outward field extends over the double Lemma
• Boundary orientation is independent of the outward vector field Proposition
• Boundary-defining functions exist locally and detect inward vectors Proposition
• The boundary tangent space is the boundary-tangent hyperplane Proposition
• The outward boundary hypothesis cannot be replaced by nonzero on the boundary Remark
• A global inward-pointing boundary vector field exists Theorem
• Inward-pointing fields have local forward semiflows at the boundary Theorem
• Poincare-Hopf with outward-pointing boundary Theorem
• Positive oriented atlases characterize orientations except for one-manifolds with boundary Theorem

Dependency tree · two levels

12 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• Ioan Mărcuț, Manifolds (2017 lecture notes), §§14.5, 15.1 (standard reference, not scraped)
• Will Merry, Differential Geometry (2021), Lecture 24 (standard reference, not scraped)