Alphabeta Math
PropositionStatement: AI-adaptedProof: AI-adaptedPipeline-generated
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The riemannian hessian is symmetric

Statement

For a smooth function f on a Riemannian manifold, the Levi–Civita Hessian is symmetric: Hess⁡f(X,Y)=Hess⁡f(Y,X) for all local smooth vector fields.

Facts & Assumptions

Given: A smooth Riemannian metric and smooth f.

[F1]

Hess⁡f(X,Y)=X(Yf)−(∇XY)f is a smooth two-tensor (Gradient hessian and divergence connection formulas).

[F2]

Levi–Civita exists and is torsion free (Fundamental theorem of riemannian geometry).

[F3]

[X,Y]f=X(Yf)−Y(Xf) (The Lie bracket of smooth vector fields).

Proof

1.1

Subtract the two formulas in [F1]. By [F3], the difference is [X,Y]f−(∇XY−∇YX)f. Torsion freeness in [F2] identifies the vector fields in the parentheses with [X,Y], so the difference is zero. This uses no assumption that X,Y commute.

F1F2F3
2.1

The identity holds for all local fields, hence all tangent vectors by the tensoriality in [F1]. Zero fields and constant functions give zero values; dimension zero has the zero tensor and dimension one the single symmetric entry. The same bracket identity and torsion equation apply in boundary charts, so no endpoint exception or choice assumption is introduced.

F1step 1.1∎

Depends on

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