Page 85 - Intro to Tensor Calculus

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Riemann Space V N
A Riemannian space V N is said to exist if the element of arc length squared has the form

2
i
ds = g ij dx dx j (1.3.23)
1
N
2
where the metrices g ij = g ij (x ,x ,...,x ) are continuous functions of the coordinates and are different
from constants. In the special case g ij = δ ij the Riemannian space V N reduces to a Euclidean space E N .
The element of arc length squared defined by equation (1.3.23) is called the Riemannian metric and any
geometry which results by using this metric is called a Riemannian geometry. A space V N is called flat if
2
i 2
it is possible to find a coordinate transformation where the element of arclength squared is ds = i (dx )
where each i is either +1 or −1. A space which is not flat is called curved.

Geometry in V N
j ~
i ~
~
~
Given two vectors A = A E i and B = B E j , then their dot product can be represented
ij
i
j
~
j
i
i
j ~
~ ~
~ ~
A · B = A B E i · E j = g ij A B = A j B = A B i = g A j B i = |A||B| cos θ. (1.3.24)
~
~
Consequently, in an N dimensional Riemannian space V N the dot or inner product of two vectors A and B
is defined:
j
i
i
ij
j
g ij A B = A j B = A B i = g A j B i = AB cos θ. (1.3.25)
i
In this definition A is the magnitude of the vector A , the quantity B is the magnitude of the vector B i and
θ is the angle between the vectors when their origins are made to coincide. In the special case that θ =90 ◦
i
i
i
j
we have g ij A B = 0 as the condition that must be satisfied in order that the given vectors A and B are
i
i
orthogonal to one another. Consider also the special case of equation (1.3.25) when A = B and θ =0. In
this case the equations (1.3.25) inform us that
2
i
n
i
in
g A n A i = A A i = g in A A =(A) . (1.3.26)
i
From this equation one can determine the magnitude of the vector A . The magnitudes A and B can be
1 1
q
p
i
n
written A =(g in A A ) 2 and B =(g pq B B ) 2 and so we can express equation (1.3.24) in the form
i
g ij A B j
cos θ = 1 1 . (1.3.27)
q
p
n
m
(g mn A A ) 2 (g pq B B ) 2
An import application of the above concepts arises in the dynamics of rigid body motion. Note that if a
i
i
vector A has constant magnitude and the magnitude of dA i is different from zero, then the vectors A and
dt
dA i must be orthogonal to one another due to the fact that g ij A i dA j =0. As an example, consider the unit
dt dt
vectors b e 1 , b e 2 and b e 3 on a rotating system of Cartesian axes. We have for constants c i , i =1, 6that
d b e 1 d b e 2 d b e 3
= c 1 b e 2 + c 2 b e 3 = c 3 b e 3 + c 4 b e 1 = c 5 b e 1 + c 6 b e 2
dt dt dt
because the derivative of any b e i (i fixed) constant vector must lie in a plane containing the vectors b e j and
b e k ,(j 6= i , k 6= i and j 6= k), since any vector in this plane must be perpendicular to b e i .