Riemannian Metrics with the Prescribed Curvature Tensor and all Its Covariant Derivatives at One Point

Abstract

Abstract On an n ‐dimensional vector space, equipped with a scalar product, we prescribe (0, 4) ‐, (0, 5)‐, … type tensors R (0) , R (1) , …, satisfying the well‐known conditions for a curvature tensor and its derivatives and furthermore certain inequalities for the absolute values of the components of R (k) . Then there is an analytic Riemannian metric g on an open ball of the Cartesian space R n [ u 1 , …, u n ] for which u 1 , …, u n are normal coordinates and (▽ (k) R ) 0 = R (k) ( k = 0, 1, 2, …) hold under an identification of the tangent space T 0 R n at the origin with the vector space; ▽ (k) R denote the curvature tensor and its covariant derivatives with respect to the Levi‐Civita connection ▽ of g, respectively.