Page 152 - Intro to Tensor Calculus

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(see example 1.5-1) which is an invariant of the surface and called the Gaussian curvature or total curvature.
In the exercises following this section it is shown that the Riemann Christoffel tensor of the surface can be
expressed in terms of the total curvature and the alternating tensors as

R αβγδ = K αβ γδ . (1.5.94)

r
Consider the second tensor derivative of x which is given by
α
∂x r α,β r δ δ
x r α,βγ = + x r α,β γ n x r δ,β − x r α,γ (1.5.95)
x −
∂u γ mn αγ βγ
g a a
which can be shown to satisfy the relation

x r α,βγ − x r α,γβ = R δ .αβγ δ r (1.5.96)
x .

Using the relation (1.5.96) we can now derive some interesting properties relating to the tensors a αβ ,b αβ ,
c αβ , R αβγδ , the mean curvature H and the total curvature K.
Consider the tensor derivative of the equation (1.5.71) which can be written

i
x i α,βγ = b αβ,γ n + b αβ n i ,γ (1.5.97)

where

∂b αβ σ σ
b αβ,γ = − b σβ − b ασ . (1.5.98)
∂u α αγ βγ
a a
By using the Weingarten formula, given in equation (1.5.80), the equation (1.5.97) can be expressed in the
form
i
τσ
x i α,βγ = b αβ,γ n − b αβ a b τγ x i σ (1.5.99)
and by using the equations (1.5.98) and (1.5.99) it can be established that

τδ
r
r
x r − x r =(b αβ,γ − b αγ,β )n − a (b αβ b τγ − b αγ b τβ )x . (1.5.100)
α,βγ α,γβ δ
Now by equating the results from the equations (1.5.96) and (1.5.100) we arrive at the relation
x =(b αβ,γ − b αγ,β )n − a (b αβ b τγ − b αγ b τβ )x .
R δ .αβγ δ r r τδ r δ (1.5.101)
Multiplying the equation (1.5.101) by n r and using the results from the equation (1.5.76) there results the
Codazzi equations
b αβ,γ − b αγ,β =0. (1.5.102)

Multiplying the equation (1.5.101) by g rmx m and simplifying one can derive the Gauss equations of the
σ
surface
R σαβγ = b αγ b σβ − b αβ b σγ . (1.5.103)

By using the Gauss equations (1.5.103) the equation (1.5.94) can be written as

K σα βγ = b αγ b σβ − b αβ b σγ . (1.5.104)