PropositionStatement: AI-adaptedProof: AI-adaptedPipeline-generated

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.

Levi civita connection commutes with musical isomorphisms

Statement

For Levi–Civita and its dual connection, $(\nabla_{X} α)^{♯} = \nabla_{X} (α^{♯}), \nabla_{X} (Y^{♭}) = (\nabla_{X} Y)^{♭} .$

Facts & Assumptions

Given: A Riemannian metric, its Levi–Civita connection, a one-form $α$ and fields $X, Y$ .

[F1]

Levi–Civita is metric compatible (Fundamental theorem of riemannian geometry).

[F2]

The dual derivative satisfies $(\nabla_{X} α) (Y) = X (α (Y)) - α (\nabla_{X} Y)$ (Dual connection).

[F3]

Musical maps are smooth inverse maps characterized by metric pairing (The musical maps are smooth inverse bundle isomorphisms).

Proof

1.1

Put $A = α^{♯}$ . Then $(\nabla_{X} α) (Y) = X g (A, Y) - g (A, \nabla_{X} Y) = g (\nabla_{X} A, Y)$ by compatibility. Since this holds for all local $Y$ , nondegeneracy identifies $(\nabla_{X} α)^{♯}$ with $\nabla_{X} A$ .

F1F2F3

2.1

Substitute $α = Y^{♭}$ in step 1.1 and use both inverse identities of [F3] to get the flat formula. Zero fields/forms give zero on both sides; dimension zero has the unique fibre maps and rank one requires no modification. All constructions are local and smooth at boundary points, and no basis family or metric existence theorem is used.

F3step 1.1∎

Depends on

Used by

Gradient hessian and divergence connection formulas Proposition

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

Ved Datar, Lectures on Riemannian Geometry (standard reference, not scraped)