It is shown that Weyl's geometry and an apparently similar geometry suggested by Lyra are special cases of manifolds with more general connections. The difference between the two geometries and their relationship with Riemannian geometry are clarified by giving a global formulation of Lyra's geometry. Finally the outline of a field theory based on the latter geometry is given.
REFERENCES
1.
H. Weyl, S.‐B. Preuss. Akad. Wiss., Berlin, 465 (1918).
2.
3.
4.
D. K. Sen, thesis (University of Paris, 1958).
5.
N. J. Hicks, Notes on Differential Geometry, Van Nostrand Mathematical Studies ♯3 (Van Nostrand, Princeton, N.J., 1965), p. 104.
6.
It is interesting to note that Schrödinger [Space‐Time Structure (Cambridge U.P., Cambridge, 1954), p. 6] considered the case of a nonsymmetric but metric preserving connection (in local coordinates) and showed that the antisymmetric part determines uniquely as follows: .
7.
Note that Lyra set so that, from (3.10), Lyra’s should not be confused with the components of ∇ in a local coordinate system, i.e., a local reference system with the “normal” gauge
8.
9.
D. K. Sen, Fields mid/or Particles (Academic, London, 1968), p. 84.
10.
K. A. Dunn, thesis (University of Toronto, 1971).
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© 1972 The American Institute of Physics.
1972
The American Institute of Physics
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