Questions: Orientation

Question 1

A smooth n-manifold M is orientable if and only if it admits a nowhere-vanishing n-form (volume form). What fails on a non-orientable manifold?

• A. There are no n-forms at all on a non-orientable manifold
• B. Every n-form on a non-orientable manifold must vanish somewhere
• C. Non-orientable manifolds cannot be smooth
• D. The tangent bundle of a non-orientable manifold is trivial

Answer: (B)

On a non-orientable manifold, n-forms exist but every globally defined n-form must vanish at some point. This is because a nowhere-vanishing n-form would define an orientation (positive at every point), which is impossible on a non-orientable manifold. The Mobius band, for example, has plenty of 2-forms defined on patches, but no single 2-form that is nonzero everywhere. The vanishing is forced by the topological twist that prevents a consistent orientation.

Question 2

The sphere S² is orientable, but the real projective plane ℝP² is not.

• T. True
• F. False

Answer: True

S² is orientable — the area form inherited from its embedding in ℝ³ (or equivalently, the outward unit normal) provides a consistent orientation. ℝP² is obtained from S² by identifying antipodal points, and this identification reverses orientation (the antipodal map on S² has degree -1 for even-dimensional spheres). Since the quotient map reverses orientation, ℝP² is non-orientable. More generally, ℝPⁿ is orientable if and only if n is odd.

Question 3

Why is orientation required for integration of differential forms? What goes wrong without it?

Model answer: Integration of an n-form on an n-manifold involves choosing local coordinates, computing the integral in coordinates, and patching together via a partition of unity. The change-of-variables formula introduces a factor of |det(Jacobian)|, but differential forms naturally transform by det(Jacobian) without the absolute value. If the manifold has charts with both positive and negative Jacobian determinants (incompatible orientations), the contributions from different charts can cancel instead of adding up, giving inconsistent results. An orientation ensures all transition maps have positive Jacobian determinant, making the signs consistent.

On an oriented manifold, the integral ∫_M ω is well-defined and changes sign if you reverse the orientation. On a non-orientable manifold, you can integrate densities (which transform by |det J|) but not differential forms. This is why Stokes' theorem requires oriented manifolds — the boundary orientation must be compatible with the interior orientation.

Question 4

An orientation on a connected manifold is determined by the orientation at a single point.

• T. True
• F. False

Answer: True

On a connected manifold, once you choose an orientation at one point (one of the two equivalence classes of ordered bases), there is at most one way to extend it continuously to the entire manifold. If the manifold is orientable, this extension exists and is unique. If not orientable, no continuous extension exists. So a connected orientable manifold has exactly two orientations (related by flipping the sign everywhere), and the choice at any single point determines the whole orientation.