Hostname: page-component-7bb8b95d7b-nptnm Total loading time: 0 Render date: 2024-10-06T23:20:16.615Z Has data issue: false hasContentIssue false
  • English
  • Français

Connections Satisfying a Generalized Ricci Lemma

Published online by Cambridge University Press:  20 November 2018

J. R. Vanstone
J. R. Vanstone*
Affiliation:
University of Toronto and Summer Research Institute of the Canadian Mathematical Congress
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we shall consider a generalization of a very old problem in differential geometry; namely, given a second-order covariant tensor field aij(x) on an n-dimensional manifold, when does there exist a connection such that the covariant derivative, defined by

vanishes?

The earliest question of this type arose in the case when is symmetric and positive definite. A solution connection of the problem is then given by the Christoffel symbols

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 16 , 1964 , pp. 549 - 560
Copyright
Copyright © Canadian Mathematical Society 1964

References

1. Eisenhart, L. P., Symmetric tensors of second order whose first covariant derivatives are zero, Trans. Am. Math. Soc, 25 (1923), 297–306.Google Scholar
2. Gantmacher, F. R., The theory of matrices, Vol. 1 (New York, 1960).Google Scholar
3. Hlavaty, V., Geometry of Einstein's Unified Field Theory (Groningen, 1957).Google Scholar
4. Van der Waerden, B. L., Modern algebra, Vol. 1 (New York, 1949).Google Scholar