feat: add Koszul formula eval problem

LeanEval/Geometry/KoszulFormula.lean

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+import Mathlib
+import EvalTools.Markers
+
+namespace LeanEval
+namespace Geometry
+
+/-!
+# Koszul formula
+
+§38 of Oliver Knill's *Some Fundamental Theorems in Mathematics* (an
+additional statement of the section on Riemannian geometry; the boxed
+main theorem is the Levi-Civita fundamental theorem). For any smooth
+torsion-free metric-compatible covariant derivative `cov` on the tangent
+bundle of a Riemannian manifold, the inner product `⟨∇_X Y, Z⟩` is
+determined by the metric and Lie brackets via
+
+`2 ⟨∇_X Y, Z⟩ = X·⟨Y, Z⟩ + Y·⟨X, Z⟩ − Z·⟨X, Y⟩`
+`               − ⟨X, [Y, Z]⟩ − ⟨Y, [X, Z]⟩ + ⟨Z, [X, Y]⟩`.
+
+This is the **Koszul formula** — the explicit identity that *forces*
+uniqueness of the Levi-Civita connection.
+
+mathlib has the covariant-derivative / Riemannian-bundle / Lie-bracket
+machinery but no metric-compatibility predicate and no Koszul formula
+(`grep -ri Koszul`: no relevant hits). One helper definition
+(`IsMetricCompatible`, ~½ page) is added here.
+-/
+
+open scoped Manifold ContDiff Bundle Topology
+open Bundle ContDiff Set VectorField CovariantDerivative
+
+/-- A covariant derivative `cov` on `TM` is **compatible with the
+Riemannian metric** if `v · ⟨Y, Z⟩ = ⟨∇_v Y, Z⟩ + ⟨Y, ∇_v Z⟩` for all
+smooth vector fields `Y, Z` and every point/direction `(x, v)`. -/
+def IsMetricCompatible
+    {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+      [FiniteDimensional ℝ E] [CompleteSpace E]
+    {H : Type*} [TopologicalSpace H] {I : ModelWithCorners ℝ E H}
+    {M : Type*} [TopologicalSpace M] [ChartedSpace H M]
+      [IsManifold I ∞ M]
+    [RiemannianBundle (fun (x : M) ↦ TangentSpace I x)]
+    [IsContMDiffRiemannianBundle I ∞ E (fun (x : M) ↦ TangentSpace I x)]
+    (cov : CovariantDerivative I E (TangentSpace I (M := M))) : Prop :=
+  ∀ (Y Z : Π x : M, TangentSpace I x),
+    CMDiff ∞ (T% Y) → CMDiff ∞ (T% Z) →
+    ∀ (x : M) (v : TangentSpace I x),
+      extDerivFun (fun y : M => inner ℝ (Y y) (Z y)) x v =
+        inner ℝ (cov Y x v) (Z x) + inner ℝ (Y x) (cov Z x v)
+
+/-- **Koszul formula.** For any smooth torsion-free metric-compatible
+covariant derivative `cov` on `TM`, `2 ⟨∇_X Y, Z⟩` equals the cyclic sum
+of directional derivatives `X·⟨Y, Z⟩ + Y·⟨X, Z⟩ − Z·⟨X, Y⟩` minus the
+Lie-bracket cyclic sum `⟨X, [Y, Z]⟩ + ⟨Y, [X, Z]⟩ − ⟨Z, [X, Y]⟩`. -/
+@[eval_problem]
+theorem koszul_formula
+    {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+      [FiniteDimensional ℝ E] [CompleteSpace E]
+    {H : Type*} [TopologicalSpace H] {I : ModelWithCorners ℝ E H}
+    {M : Type*} [TopologicalSpace M] [ChartedSpace H M]
+      [IsManifold I ∞ M]
+    [RiemannianBundle (fun (x : M) ↦ TangentSpace I x)]
+    [IsContMDiffRiemannianBundle I ∞ E (fun (x : M) ↦ TangentSpace I x)]
+    (cov : CovariantDerivative I E (TangentSpace I (M := M)))
+    [ContMDiffCovariantDerivative cov ∞]
+    (_htor : cov.torsion = 0) (_hmet : IsMetricCompatible cov)
+    (X Y Z : Π x : M, TangentSpace I x)
+    (_hX : CMDiff ∞ (T% X)) (_hY : CMDiff ∞ (T% Y)) (_hZ : CMDiff ∞ (T% Z))
+    (x : M) :
+    2 * inner ℝ (cov Y x (X x)) (Z x) =
+      extDerivFun (fun y : M => inner ℝ (Y y) (Z y)) x (X x)
+      + extDerivFun (fun y : M => inner ℝ (X y) (Z y)) x (Y x)
+      - extDerivFun (fun y : M => inner ℝ (X y) (Y y)) x (Z x)
+      - inner ℝ (X x) (mlieBracket I Y Z x)
+      - inner ℝ (Y x) (mlieBracket I X Z x)
+      + inner ℝ (Z x) (mlieBracket I X Y x) := by
+  sorry
+
+end Geometry
+end LeanEval

manifests/problems/koszul_formula.toml

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+id = "koszul_formula"
+title = "Koszul formula"
+test = false
+module = "LeanEval.Geometry.KoszulFormula"
+holes = ["koszul_formula"]
+submitter = "Kim Morrison"
+notes = "§38 of Knill's 'Some Fundamental Theorems in Mathematics' (additional statement of the Riemannian-geometry section; the boxed main theorem is Levi-Civita). The Koszul formula: 2⟨∇_X Y, Z⟩ equals the cyclic sum of directional derivatives X·⟨Y,Z⟩ + Y·⟨X,Z⟩ − Z·⟨X,Y⟩ minus the Lie-bracket cyclic sum ⟨X,[Y,Z]⟩ + ⟨Y,[X,Z]⟩ − ⟨Z,[X,Y]⟩. The identity that forces uniqueness of the Levi-Civita connection. Mathlib has the covariant-derivative / Riemannian-bundle / Lie-bracket machinery but no metric-compatibility predicate and no Koszul formula. The Challenge ships one helper (IsMetricCompatible)."
+source = "J.-L. Koszul (course notes, 1950s); cf. do Carmo, Riemannian Geometry. Listed as §38 additional statement in O. Knill, Some Fundamental Theorems in Mathematics (https://people.math.harvard.edu/~knill/graphgeometry/papers/fundamental.pdf)."
+informal_solution = "Direct algebraic manipulation from metric compatibility and torsion-freeness. Metric compatibility gives X·g(Y,Z) = g(∇_X Y, Z) + g(Y, ∇_X Z), and cyclic permutations Y·g(Z,X) and Z·g(X,Y). Torsion-freeness gives ∇_X Y − ∇_Y X = [X,Y] (and permutations). Form the linear combination X·g(Y,Z) + Y·g(X,Z) − Z·g(X,Y); expand each term via metric compatibility; substitute the torsion-free identities to convert ∇_X Z − ∇_Z X, ∇_Y Z − ∇_Z Y, and ∇_X Y − ∇_Y X into Lie brackets. The cross-terms ±g(∇_Y X, Z), ±g(∇_X Z, Y), ±g(∇_Y Z, X) collapse pairwise except for 2·g(∇_X Y, Z), yielding the formula."