Alphabeta Math
PropositionStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-14
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.

Curvature is skew in its first two arguments

Statement

For every affine connection and all smooth vector fields X,Y,Z,

R(X,Y)Z=−R(Y,X)Z.

Thus curvature is alternating in its first two arguments; no metric or torsion hypothesis is needed.

Facts & Assumptions

[F1]

Curvature is the bracket-corrected commutator R(X,Y)Z=∇X∇YZ−∇Y∇XZ−∇[X,Y]Z. Curvature of an affine connection.

Proof

Given: Smooth vector fields X,Y,Z and an affine connection ∇.

1.1

Interchanging X and Y in [F1] gives R(Y,X)Z=∇Y∇XZ−∇X∇YZ−∇[Y,X]Z.

F1
2.1

The Lie bracket is a commutator of derivations, so [Y,X]=−[X,Y]; the connection is real-linear in its differentiating field. Hence the expression in step 1.1 is −(∇X∇YZ−∇Y∇XZ−∇[X,Y]Z)=−R(X,Y)Z.

F1step 1.1algebra∎

Depends on

Used by

Dependency tree · two levels

3 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