Mathematical Principles of Theoretical Physics

3.1 Basic Concepts

curves and two-dimensional surfaces. It is, however, difficult for us to tell whether the three-
dimensional space we live in is curved or flat by our common sense. The Riemannian geom-
etry provides a theory with which the intelligent beings living in ann-dimensional manifold
are able to determine the curvature of this space from its metric.
A plane in Figure3.1(a) is expressed by

(3.1.1) ~r(x^1 ,x^2 ) = (x^1 ,x^2 ,x^3 (x^1 ,x^2 )),

wherex 3 =α 1 x 1 +α 2 x 2 +α 3 ,αj( 1 ≤j≤ 3 )are constants, and the metric of the plane (3.1.1)
are given by

(3.1.2) ds^2 =d~r·d~r=(dx^1 )^2 + (dx^2 )^2 + (α 1 dx^1 +α 2 dx^2 )^2

=( 1 +α 12 )(dx^1 )^2 + 2 α 1 α 2 dx^1 dx^2 + ( 1 +α 22 )(dx^2 )^2.

A sphere in Figure3.1(b) is expressed by

(3.1.3) ~r(x^1 ,x^2 ) = (x^1 ,x^2 ,x^3 (x^1 ,x^2 )),

wherex^3 =

√

R^2 −(x^1 )^2 −(x^2 )^2 , andRis the radius. The metric of this sphere (3.1.3) is
given by

ds^2 =(dx^1 )^2 + (dx^2 )^2 +

(

∂x^3

∂x^1

dx^1 +

∂x^3

∂x^2

dx^2

) 2

(3.1.4)

=( 1 +φ)(dx^1 )^2 + 2 φdx^1 dx^2 + ( 1 +φ)(dx^2 )^2 ,

whereφ= 1 /(R^2 −(x^1 )^2 −(x^2 )^2 ).

x^1

x^2

x^3

⇀r

(x^1 , x^2 )

(a)

x^1

x^3

x^2

⇀r

(x^1 , x^2 )

(b)

Figure 3.1: (a) a plane, and (b) a sphere with radiusR.

It is clear that the metric

ds^2 =gijdxidxj

defined on a surface dictates its curvature. We see that the metric (3.1.2) of the plane (3.1.1),
i.e.

(3.1.5)

(

g 11 g 12

g 21 g 22

)

=

(

1 +α 12 α 1 α 2

α 1 α 2 1 +α^22

)