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.

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H. Weyl, S.‐B. Preuss. Akad. Wiss., Berlin, 465 (1918).
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G.
Folland
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4
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G.
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4.
D. K. Sen, thesis (University of Paris, 1958).
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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 Γαβμ (in local coordinates) and showed that the antisymmetric part Γαβμ determines Γαβμ uniquely as follows:
.
7.
Note that Lyra set Γαβμ = Γαβμ12δαμφβ so that, from (3.10), Γαβμ = (1/x0){μαβ}+12αμβμφα−gαβφμ). Lyra’s Γαβμ should not be confused with the components of ∇ in a local coordinate system, i.e., a local reference system with the “normal” gauge x0 = 1.
8.
D. K.
Sen
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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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