Contracting a Kulkarni–Nomizu product with the inner product

On a finite-dimensional real inner product space W of dimension n, contracting the first and fourth arguments of the Kulkarni–Nomizu product h ⊙ g of a bilinear form h with the inner product g gives

∑ᵢ (h ⊙ g)(eᵢ, y, z, eᵢ) = (n - 2) h(y, z) + (tr_g h) g(y, z)

for every orthonormal basis (eᵢ). This is the Ricci-type contraction of h ⊙ g; it is the identity that determines the Schouten tensor of a Riemannian metric as the unique bilinear form P with Ric = (n - 2) P + (tr_g P) g, and that makes the Weyl tensor trace-free.

Taking the trace once more shows that in dimension at least 3 the map h ↦ h ⊙ g is injective (LinearMap.BilinForm.kulkarniNomizu_innerₗ_eq_zero_iff): a bilinear form whose Kulkarni–Nomizu product with the inner product vanishes is zero. In dimension 2 this fails, since there (h ⊙ g) only sees the trace of h.

References

• A. L. Besse, Einstein Manifolds, Springer (1987), Section 1.G.

theorem LinearMap.BilinForm.sum_kulkarniNomizu_innerₗ {W : Type u_1} [NormedAddCommGroup W] [InnerProductSpace ℝ W] [FiniteDimensional ℝ W] {ι : Type u_2} [Fintype ι] (h : LinearMap.BilinForm ℝ W) (b : OrthonormalBasis ι ℝ W) (y z : W) : ∑ j : ι, (((((kulkarniNomizu h) (innerₗ W)) (b j)) y) z) (b j) = (↑(Module.finrank ℝ W) - 2) * (h y) z + TauCeti.bilinFormTrace h * inner ℝ y z

In an orthonormal basis, contracting the first and fourth arguments of the Kulkarni–Nomizu product of h with the inner product gives (n - 2) h + (tr h) g.

@[simp]

theorem LinearMap.BilinForm.kulkarniNomizu_innerₗ_eq_zero_iff {W : Type u_1} [NormedAddCommGroup W] [InnerProductSpace ℝ W] [FiniteDimensional ℝ W] (hW : 3 ≤ Module.finrank ℝ W) {h : LinearMap.BilinForm ℝ W} : (kulkarniNomizu h) (innerₗ W) = 0 ↔ h = 0

In dimension at least 3, the Kulkarni–Nomizu product with the inner product is injective: h ⊙ g = 0 only for h = 0.