, 
) 
R{sub
} = g{sub
} R{sub
}
.
Conversely, R{sub
}
= g{sup
} (
,
). Like equation (95),
the symbols are antisymmetric in
, . Three more
linear relations hold between them. This reduces the number
of independent symbols to n
(n
- 1)/12--e.g., six for a three-space, 20 for an S
, and but
one for a surface, say (12, 12). This symbol divided by g
is a scalar of the surface, its
Gaussian curvature K.
If a vector undergoes parallel displacement around a circuit
on a surface, it returns with its direction changed by an amount
equal to 
Kd
, the Gaussian curvature of the surface integrated over
the surface area enclosed by the circuit. As a consequence,
the excess (over 
)
of the angle sum in any geodesic triangle is
Kd, a theorem
proved by Gauss.
When there are three or more dimensions, the curvature properties
can no longer be expressed by a single magnitude but require
for their description the knowledge of the whole curvature tensor
or the associated Riemann symbols. The concept of Gaussian curvature
is now replaced by that of
Riemannian curvature--i.e.,
at any point O(x) of S
, the set of Gaussian curvatures K of geodesic
surfaces of all possible orientations laid through O.
If h{sub
} be the metric
tensor of such a surface as submanifold, K = 1/h(12,
12)
,
the symbol is calculated with h{sub
}. This and the
determinant h can be expressed in terms of the tensor
g{sub
} of the manifold and the vector pair, 
{sup },
{sup
}, fixing the orientation (see 96).
In general, K will depend on position and on orientation.
In other words, with regard to curvature, S
may be nonhomogeneous and anisotropic (e.g., space-time
within or around matter); but, if K is everywhere isotropic,
it is also constant throughout S
(Schur's theorem). By the expression (96)
for K in terms of the tensor g{sub
} and the vector
pair, {sup
}, {sup
}, the
necessary and sufficient condition for isotropy of Riemannian
curvature becomes (
,
) = K(g{sub
} g{sub
} - g{sub
} g{sub
}) or R{sub
}
= K(
g{sub
} -
g{sub
}) with constant K. (
D.T.F./Ed.)
