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By Chern-Weil theory, the forms ∆sF ,g (hj ) = hj (∇g , s) restricted to Lx = SO(q) are closed, and their classes in cohomology define the free exterior generators for the cohomology H ∗ (SO(q); R). ) Give the algebra RWq the basic filtration by the degree in I(SO(q))q , and the forms in Ω∗ (M ) the basic filtration by their degree in π ∗ Ω∗ (B). ) The characteristic map ∆sF ,g preserves the filtrations, hence induces a map of the associated Leray-Hirsch spectral sequences, ∗,∗ ∗,∗ ∆∗,∗ r : Er (RWq , dW ) → Er (M, dr ) For r = 2, we then have ∆∗,∗ : E ∗,∗ (RWq ) ∼ = (RWq , dW ) → E ∗,∗ (M, d2 ) ∼ = H ∗ (SO(q); R)⊗H ∗ (B; R) 2 2 r which is injective by the remark above.

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A Beckman-Quarles type theorem for finite desarguesian planes by Benz W.

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