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Curvature homogeneity for four-dimensional manifolds
Sekigawa, Kouei,Suga, Hiroshi,Vanhecke, Lieven Korean Mathematical Society 1995 대한수학회지 Vol.32 No.1
Let (M,g) be an n-dimensional, connected Riemannian manifold with Levi Civita connection $\nabla$ and Riemannian curvature tensor R defined by $$ R_XY = [\nabla_X, \nabla_Y] - \nabla_{[X,Y]} $$ for all smooth vector fields X, Y. $\nablaR, \cdots, \nabla^kR, \cdots$ denote the successive covariant derivatives and we assume $\nabla^0R = R$.