A static cosmological model of the universe based on LYRA'$ modified Riemannian Geometry is proposed. The red-shift of spectral lines from extra-galactic nebulae is shown to be a consequence of an inherent geometrical property of the model independent O f expansion. The model is similar to the static EINSTEIN model but shows a red-shift and has a finite density even without the introduction of a cosmological constant.Introduction. The discovery of Hubble-effect [1] seems to have firmly established the idea of an expanding universe in cosmological theories. This is of course based on the usual interpretation of red-shift as a Doppler effect. Vghen velocities of nebulae, as such determined were reasonably small enough nobody thought of any alternative explanation of the phenomenon of red-shift. But when distant nebulae were found to have enormous velocities of the order of seventh or fifth that of light some people began to think about alternative explanations of red-shift. The attempts of Zwicky to explain red-shift as a reddening due to gravitational fields of the galaxies or the possibility of a Compton-effect

A new scalar-tensor theory of gravitation is formulated in a modified Riemannian manifold in which both the scalar and tensor fields have intrinsic geometrical significance. This is in contrast to the well-known Brans-Dicke theory where the tensor field alone is geometrized and the scalar field is alien to the geometry. The static spherically symmetric solution of the exterior field equations is worked out in detail.

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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