Page 173 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes Formula (10) is called the recovery formula. It recovers bilinear function a from quadratic
function f produced in (8). Due to this formula, in referring to a quadratic form we always imply
some symmetric bilinear form like the geometric tensorial object introduced by definition.

15.5 General Definition of Tensors

Vectors, covectors, linear operators, and bilinear forms are examples of tensors. They are
geometric objects that are represented numerically when some basis in the space is chosen. This
numeric representation is specific to each of them: vectors and covectors are represented by one-
dimensional arrays, linear operators and quadratic forms are represented by two-dimensional
arrays. Apart from the number of indices, their position does matter. The coordinates of a vector
are numerated by one upper index, which is called the contra-variant index. The coordinates of
a covector are numerated by one lower index, which is called the covariant index. In a matrix of
bilinear form we use two lower indices; therefore bilinear forms are called twice-covariant
tensors. Linear operators are tensors of mixed type; their components are numerated by one
upper and one lower index. The number of indices and their positions determine the
transformation rules, i.e. the way the components of each particular tensor behave under a
change of basis. In the general case, any tensor is represented by a multidimensional array with
a definite number of upper indices and a definite number of lower indices.

Let’s denote these numbers by r and s. Then we have a tensor of the type (r; s), or sometimes the
term valency is used. A tensor of type (r; s), or of valency (r; s) is called an r-times contravariant
and an s-times covariant tensor. This is terminology; now let’s proceed to the exact definition. It
is based on the following general transformation formulas:

3 3
r i
X 1 i ...i r ... S ...S T ...T X h k 1 ...h r , ...(1)
1 i
k
s k 
1 j ...j =  
1
1 j
s j
1 ...k
r h
h
s
h 1 , ..., r h 1 s
k 1 , ..., s k
3 3
1 ...h
1 i
X 1 j ...j =   T ...T S ...S X h k 1 ...k r , ...(2)
r i
...
k
s k
1 i ...i

1
r
1 j
s j
h
r h
s
h 1 , ..., r h 1 s
k 1 , ..., s k
Definition: A geometric object X in each basis represented by (r + s) dimensional array X 1 i ...i s r
1 j ...j of
real numbers and such that the components of this array obey the transformation rules (1) and
(2) under a change of basis is called tensor of type (r, s), or of valency (r, s).
Formula (2) is derived from (1), so it is sufficient to remember only one of them. Let it be the
formula (1). Though huge, formula (1) is easy to remember.
Indices i , ... i and j , ..., j are free indices. In right hand side of the equality (1) they are distributed
1
s
1
r
in S-s and T-s, each having only one entry and each keeping its position, i.e. upper indices i , ...;
1
i remain upper and lower indices j ,..., j remain lower in right hand side of the equality (1).
s
r
1
Other indices h , ... h and k , ... , k are summation indices; they enter the right hand side of (1)
r
s
1
1
pairwise: once as an upper index and once as a lower index, once in S-s or T-s and once in
components of array X h  k 1 ...h r s .
1 ...k
When expressing X 1 i ...i s r h  k 1 ...k r s each upper index is served by direct transition matrix
1 ...h
1 j ...j through X
S and produces one summation in (1):
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