Riemannian Metrics with the Prescribed Curvature Tensor and all Its Covariant Derivatives at One Point
✍ Scribed by Oldřich Kowalski; Martin Belger
- Publisher
- John Wiley and Sons
- Year
- 2006
- Tongue
- English
- Weight
- 697 KB
- Volume
- 168
- Category
- Article
- ISSN
- 0025-584X
No coin nor oath required. For personal study only.
✦ Synopsis
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.