Page 62 - Curvature and Homology

44 I. RIEMANNIAN MANIFOLDS

2. If 0, + denote polar coordinates on a sphere in ES the manifold can be covered
by the neighborhoods

with coordinates
0 0
u1 = tan - cos 4, uB = - tan - sin 4.
2 2
and
0 0
utl = cot - cos 4, ula = cot - sin 4,
2 2
respectively. Show that the sphere is orientable.
On the other hand, the real projective plane PZ is not an orientable manifold.
For, denoting by x, y, z rectangular cartesian coordinates in ES, P2 can be
covered by the neighborhoods:

with the corresponding coordinates

and

Incidentally, the compact surfaces can be classified as spheres or projective
planes with various numbers of handles attached.

C. Grassman algebra
1. Let E be an associative algebra over the reds R with the properties:
1) E is a graded algebra (cf. 5 3.3), that is E = Eo @ El @ ... @ En @ ...,
where the operation @ denotes the direct sum; each E, is a subspace of E and
for e, E Ei, ej E Ej, ei A ej E Ei+, where A denotes multiplication in E;
2) El = V where V is a real n-dimensional vector space and Eo = R;
3) El together with the identity 1 E R generates E;
4) x Ax=O, xeE1;
5) pxl A .., A x,, = 0, x1 A ... A x, # 0, x,, ..., xn E El implies p = 0.
Then E is isomorphic to A (V).