Page 175 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes

Notes The first four properties of the scalar product (1) are quite similar to those
or quadratic forms. This is not an occasional coincidence.

Exercise 15.20: Let’s consider two arbitrary vectors x and y expanded in some basis e , e , e . This
3
1
2
means that we have the following expressions for them:
3 3
j
i
x   x e , y   x e . ...(2)
i
j
i 1 j 1


Substitute (2) into (1) and using properties (1)-(4) listed in exercise 15.17 derive the following
formula for the scalar product of x and y:
3 3
i
(x,y)   (e ,e )x y . j ...(3)
j
i
i 1 j 1


Exercise 15.21: Denote g = (e , e) and rewrite formula (3) as
j
i
ij
3 3
i
(x,y)   g x y . j ...(4)
ij
i 1 j 1


  
(e ,e ) and prove that matrices g and g 
Consider some other basis e ,e ,e , denote g  pq   p  q ij pq are
2
1
3
components of a geometric object under a change of base. Thus you prove that the Gram matrix
g = (e , e ) ...(5)
j
i
ij
defines tensor of type (0; 2). This is very important tensor; it is called the metric tensor. It
describes not only the scalar product in form of (4), but the whole geometry of our space.
Evidences for this fact are below.
Matrix (5) is symmetric due to property (5) in task on previous page. Now, keeping in mind the
tensorial nature of matrix (5), we conclude that the scalar product is a symmetric bilinear form:
(x, y) = g(x, y) ...(6)
The quadratic form corresponding to (6) is very simple: f(x) = g(x, x) = |x| . The inverse matrix
2
for (5) is denoted by the same symbol g but with upper indices: g . It determines a tensor of type
ij
(2, 0), this tensor is called dual metric tensor.
15.7 Multiplication by Numbers and Addition
Tensor operations are used to produce new tensors from those we already have. The most
simple of them are multiplication by number and addition. If we have some tensor X of type
(r, s) and a real number , then in some base e , e , e we have the array of components of tensor
3
2
1
X; let’s denote it X 1 i ...i r s . Then by multiplying all the components of this array by  we get
1 j ...j
another array
Y 1 j ...j s r   X 1 j ....j r s . ...(1)
1 i ...i
1 i ...i

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