for any X, Y [member of] [GAMMA](TM), where S, Q and [GAMMA](TM) denote the Ricci

curvature tensor, the Ricci operator with respect to the metric g and the Lie algebra of all vector fields on [M.

The density [rho] is obtained from the

curvature tensor of (1)

alpha][beta]] of the

curvature tensor of S, and the covariant of the third fundament form on S are then defined as follows (the explicit dependence on the variable y [member of] [bar.

It is well known that the structure of projective or KEnhler manifolds is governed by positivity or negativity properties of the

curvature tensor.

n],) (n > 3) the

curvature tensor R of type (0, 4) has the following form:

where S is the Ricci tensor of type (0, 2) and R is the

curvature tensor of type (1 , 3).

His topics include the basics of geometry and relativity, affine connection and covariant derivative, the geodesic equation and its applications,

curvature tensor and Einstein's equation, black holes, and cosmological models and the big bang theory.

If the

curvature tensor R of the Riemannian manifold M satisfies

We denote by M(c), then the Riemannian

curvature tensor of M(c) is given by

n+2] and are a consequence of the Bianchi identities [1] for the

curvature tensor.

Synge's famous phrase, in General Relativity Theory the

curvature tensor "is the gravitational field," then that body is not force-free since the

curvature tensor cannot be made to vanish by a change of frames of reference.

As the shell becomes thicker, the contribution of Gauss

curvature tensor in terms of energy is no more negligible as compared to that of the first two tensors used in classical thin shells.