ExampleConstruction: Literature-sourcedVerification: AI-generatedPipeline-generatedprecheck passaudited 2026-09-07

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Positive-dimensional real projective space is orientable exactly in odd dimension

Example

For $n \geq 1$ , ${R P}^{n}$ is orientable exactly when $n$ is odd; ${R P}^{0}$ is a point and is orientable.

Facts & Assumptions

Given: An integer $n \geq 1$ , the standard sphere orientation on $S^{n} = \partial B^{n + 1}$ , the antipodal map $a : S^{n} \to S^{n}$ , $a (x) = - x$ , and the quotient covering $π : S^{n} \to {R P}^{n} = S^{n} / {1, a}$ .

[L1]

The orientation on the boundary of the standard oriented ball is outward-normal-first (Induced boundary orientation).

[L2]

Between manifolds equipped with chosen orientations, a local diffeomorphism has a well-defined pointwise orientation sign, constant on a nonempty connected source (Pointwise orientation sign of a local diffeomorphism).

[L3]

A zero-dimensional real vector space has two determinant-line orientation rays (Determinant-line orientations of finite-dimensional real vector spaces).

Verification

technique · direct

1.1givenL1L2algebra

Fix $x \in S^{n}$ and a positive tangent basis $(v_{1}, \dots, v_{n})$ at $x$ . By [L1], $(x, v_{1}, \dots, v_{n})$ is positive in $R^{n + 1}$ . Since $d a_{x} (v_{i}) = - v_{i}$ , the corresponding ambient tuple at $- x$ is $(- x, - v_{1}, \dots, - v_{n})$ , whose sign relative to the original tuple is $(- 1)^{n + 1}$ . Thus [L2] gives the antipodal map the constant orientation sign $(- 1)^{n + 1}$ .

2.1givenL2L3step 1.1∎

If $a$ preserves orientation, define the orientation ray at $[x]$ by pushing the ray at $x$ forward with $d π_{x}$ . The other lift is $a (x)$ , and $π \circ a = π$ makes the resulting ray independent of that choice. Conversely, an orientation on ${R P}^{n}$ pulls back through the local diffeomorphism $π$ to an orientation of $S^{n}$ that $a$ must preserve. By step 1.1 this occurs exactly when $(- 1)^{n + 1} = 1$ , namely when $n$ is odd. Finally, ${R P}^{0}$ is a point and is orientable by [L3].

Depends on

Used by

The real projective plane is not unoriented null-cobordant Counterexample
Boy's surface: an immersion of the real projective plane in three-space Example
Maps from even projective space to the sphere use mod-two degree Example
Projective-plane curvature via a hemisphere Example
Two projective lines have one mod 2 intersection Example

Dependency tree · two levels

11 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)