Page 13 - Curvature and Homology

The symbols used have gained general acceptance with some ex-
ceptions. In particular, R and C are the fields of real and complex
numbers, respectively. (In 5 7.1, the same letter C is employed as an
operator and should cause no confusion.) The commonly used symbols
E, V, n, g, sup, inf, are not listed. The exterior or Grassman algebra
of a vector space V (over R or C) is written as A(V). By AP(V) is
meant the vector space of its elements of degree p and A denotes
multiplication in A(V). The elements of A(V) are designated by
Greek letters. The symbol M is reserved for a topological manifold,
Tp its tangent space at a point P E M (in case M is a differentiable
manifold) and T,X the dual space (of covectors). The space of tangent
vector fields is denoted by T and its dual by T*. The Lie bracket of
tangent vectors X and Y is written as [X, Y]. Tensors are generally
denoted by Latin letters. For example, the metric tensor of a Riemann-
ian manifold will usually be denoted by g. The covariant form of X
(with respect to g) is designated by the corresponding Greek symbol 6.
The notation for composition of functions (maps) employed is flexible.
It is sometimes written as g f and at other times the dot is not present.
The dot is also used to denote the (local) scalar product of vectors
(relative to g). However, no confusion should arise.

Symbol Page
n-dimensional Euclidean space ....
n-dimensional affine space ......
An with a distinguished point .....
complex n-dimensional vector space . .
n-sphere .............
n-dimensional complex torus .....
n-dimensional complex projective space
ring of integers ...........
empty set .............
tensor space-of tensors of type (r, s) . ,
Kronecker symbol .........
inner product, local scalar product . .
global scalar product ........
Hilbert space norm .........
direct sum .............
tensor product ...........
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