Page 134 - Intro to Tensor Calculus

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§1.5 DIFFERENTIAL GEOMETRY AND RELATIVITY
In this section we will examine some fundamental properties of curves and surfaces. In particular, at
each point of a space curve we can construct a moving coordinate system consisting of a tangent vector, a
normal vector and a binormal vector which is perpendicular to both the tangent and normal vectors. How
these vectors change as we move along the space curve brings up the subjects of curvature and torsion
associated with a space curve. The curvature is a measure of how the tangent vector to the curve is changing
and the torsion is a measure of the twisting of the curve out of a plane. We will find that straight lines have
zero curvature and plane curves have zero torsion.
In a similar fashion, associated with every smooth surface there are two coordinate surface curves and
a normal surface vector through each point on the surface. The coordinate surface curves have tangent
vectors which together with the normal surface vectors create a set of basis vectors. These vectors can be
used to define such things as a two dimensional surface metric and a second order curvature tensor. The
coordinate curves have tangent vectors which together with the surface normal form a coordinate system at
each point of the surface. How these surface vectors change brings into consideration two different curvatures.
A normal curvature and a tangential curvature (geodesic curvature). How these curvatures are related to
the curvature tensor and to the Riemann Christoffel tensor, introduced in the last section, as well as other
interesting relationships between the various surface vectors and curvatures, is the subject area of differential
geometry.
Also presented in this section is a brief introduction to relativity where again the Riemann Christoffel
tensor will occur. Properties of this important tensor are developed in the exercises of this section.

Space Curves and Curvature

i
i
For x = x (s),i =1, 2, 3, a 3-dimensional space curve in a Riemannian space V n with metric tensor g ij ,
i
and arc length parameter s, the vector T = dx i represents a tangent vector to the curve at a point P on
ds
i
the curve. The vector T is a unit vector because
i
dx dx j
i j
g ij T T = g ij =1. (1.5.1)
ds ds
Differentiate intrinsically, with respect to arc length, the relation (1.5.1) and verify that
δT j δT i j
i
g ij T + g ij T =0, (1.5.2)
δs δs
which implies that
δT i
j
g ij T =0. (1.5.3)
δs
i
i
Hence, the vector δT i is perpendicular to the tangent vector T . Define the unit normal vector N to the
δs
space curve to be in the same direction as the vector δT i and write
δs
1 δT i
i
N = (1.5.4)
κ δs
where κ is a scale factor, called the curvature, and is selected such that
i
δT δT j 2
i
j
g ij N N = 1 which implies g ij = κ . (1.5.5)
δs δs