110 CHAPTER 3. MATHEMATICAL FOUNDATIONS
is a constant metric, and the metric (3.1.4) of the sphere (3.1.3), i.e.
(3.1.6)
(
g 11 g 12
g 21 g 22)
=
(
1 +φ(x^1 ,x^2 ) φ(x^1 ,x^2 )
φ(x^1 ,x^2 ) 1 +φ(x^1 ,x^2 ))
is not constant. In fact, for the plane shown by Figure3.1(a), we can find a coordinate
transformation
(
̃x^1
̃x^2
)
=
(
a^11 a^12
a^21 a^22)(
x^1
x^2)
,
such that in the coordinate system( ̃x^1 , ̃x^2 ), the metric (3.1.2) is expressed in the diagonal
form:
(3.1.7) ds^2 = (d ̃x^1 )^2 + (d ̃x^2 )^2.
In other words, the metric isgij=δij. However, it is impossible to achieve this for the metric
(3.1.4) of the sphere.
The conclusion in this example holds true as well for all Riemannian manifolds. Namely,
for a Riemannian manifold{M,gij},Mis flat if and only if there is a coordinate systemx,
such that the metric{gij}can be expressed asgij=δijunder thex-coordinate system.
The following are a few general properties of ann-dimensional Riemannian manifold
{M,gij}.
- The Riemannian metric
(3.1.8) ds^2 =gijdxidxj
is invariant. In fact, under a coordinate transformation
(3.1.9) ̃x=φ(x), x=φ−^1 ( ̃x),
the second-order covariant tensor field{gij}satisfies that
(g ̃ij) = (bki)T(gkl)(blj),
d ̃x^1
..
.
d ̃xn
= (aij)
dx^1
..
.
dxn
,
(aij) =(
∂ φi
∂xj)
, (bij) = (aij)−^1.