Page 53 - Curvature and Homology

1.10. SECTIONAL CURVATURE 35

coordinates (ui) and (Bi), respectively, then it can be shown that if f
is a local differentiable homeomorphism f : U -+ 0 such that f*(&2) =
h2, then f *8, = 0, and f *oij = Oij, and conversely, if we write
= 0, @ 0,, i = 1, ..., n where @ denotes the tensor product of
covectors (cf. 1.A)
f*(e: + -.. + e,z) = q + -.. + 9;
where f* is the induced dual map. (The forms 4, i = 1, ..., n are
vectors determined by duality from the vectors e, by means of the
metric). Therefore, f induces a homeomorphism of the bundles 8,
and Bit of orthonormal frames over U and 0, respectively.
It follows that the forms 0i and Oij are intrinsically associated with the
Riemannian metric in the sense that the dual of the homeomorphism
8, -+ Do maps the 8, into the 0, and the 8,1 into the 0,, and for this
reason they account for the important properties of Riemannian
geometry.

1 .lo. Sectional curvature
In a 2-dimensional Riemannian manifold the only non-vanishing
functions Sijkl are S1212 = - Slezl = - Szl12 = S,,,,. We remark
that the Sijkr are not the components of a tensor but are, in any case,
functions defined on the bundle 8 of orthonormal frames. Moreover,
the quantity - S,,,, is the Gaussian curvature of the manifold. We
proceed to show that the value of the function - S,,,, at a point P
in an n-dimensional Riemannian manifold M is the Gaussian curvature
at P of some surface (2-dimensional submanifold) through P. To this end,
consider the family 9 of orthonormal frames {el, ..., en) at a point P
of M with the property that the 'first' two vectors of each of these frames
lie in the same plane .rr through P. Let S be a 2-dimensional submanifold
through P whose tangent plane at P is .rr. The surface S is said to be
geodesic at P if the geodesics (cf. 9 1.1 1) through P tangent to .rr all
lie on S. We seek the condition that S be geodesic at P. Let C be a
parametrized curve on S through P tangent to the vector z:,, x,e, at P
and develop the frames along C into En. If we denote the image of a
frame {el, ..., en) by {e; , ..., e;}, we have

In order that C be a geodesic, Z:,, xa 0,, must vanish, and since this
holds for arbitrary initial values of the xa, the forms Oa, (1 5 a 5 2,