We prove that energy minimizing Yang–Mills connections on compact homogeneous 4-manifolds are either instantons or split into a sum of instantons on passage to the adjoint bundle. We prove related results for Calabi–Yau 3-folds and for 3−dimensional manifolds of nonnegative Ricci curvature.

Motivated by results on the simplicial volume of locally symmetric spaces of finite volume, in this note, we observe that the simplicial volume of the moduli space Mg,n is equal to 0 if g  2; g = 1, n  3; or g = 0, n  6; and the orbifold simplicial volume of Mg,n is positive if g = 1, n = 0, 1; g = 0, n = 4. We also observe that the simplicial volume of the Deligne-Mumford compactification of Mg,n is equal to 0, and the simplicial volumes of the reductive Borel-Serre compactification of arithmetic locally symmetric spaces ..\X and the Baily-Borel compactification of Hermitian arithmetic locally symmetric spaces ..\X are also equal to 0 if the Q-rank of ..\X is at least 3 or if ..\X is irreducible and of Q-rank 2.

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We prove that if a connected Lie group action on a complete Riemannian manifold preserves the geodesics (considered as unparameterized curves), then the metric has constant positive sectional curvature, or the group acts by affine transformations.