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Riemannian Metrics with the Prescribed Curvature Tensor and all Its Covariant Derivatives at One Point
Author(s) -
Kowalski Oldřich,
Belger Martin
Publication year - 1994
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.19941680113
Subject(s) - mathematics , tangent space , riemann curvature tensor , scalar curvature , covariant transformation , curvature , mathematical analysis , curvature form , tangent vector , ricci curvature , pure mathematics , sectional curvature , mathematical physics , tangent , geometry
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.