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CHAPTER I
HISTORICAL
4. Group.
Invariance necessarily carries with it the idea of a transformation. Suppose we have a set of transformations in any variables whatever, and suppose that each of the set leaves a certain function of these variables invariant, then any transformation compounded of two or more of the set will also leave that function invariant. If any such transformation as this is not one of the original set we add it to that set, and we may thus continue adding new transformations until we reach a closed set, that is one such that if you apply in turn any two of its transformations the result is another of its transformations. Such a set is called a GROUP, and it is clear that any invariant whatever is invariant under a group of transformations.
5. In the case considered in the preceding pages there are a certain number of quadratic differential forms
together with a certain number of functions
ϕ (
x1,...,
xn), and the group of transformations
xi =
xi (
y1,...,
yn),(
i = 1,...,
n), and we suppose that under a member of this group
becomes
and that
ϕ becomes
ϕ′. Then there are deducible relations for
a′rs,
ϕ′, and their various derivatives with respect to the
y’s and for
dx1,...,
dxn in terms of the original magnitudes
ars,
ϕ, etc. In other words there exists a set of transformations for all the variables mentioned. It may be proved that this set is a group, and this group is said to be
extended from the original group. Our problem is the determination of all the invariants of this extended group.
6. Christoffel.
There have been three main methods of attack. The first, historically, is by comparison of the original and transformed forms, and in this way invariants are obtained by direct processes. The fundamental work in this direction is due to Christoffel* (1869), though the first example of an invariant, the quantity K, was given by Gauss† in 1827. Invariants which involve the derivatives of the functions are called differential parameters. Lamé‡, using the linear element in space given by ds2 = dx2 + dy2 + dz2 gave this name to the two invariants
and Beltrami § adopted it for the invariants that he discovered, those involving first and second derivatives of a function ϕ, taken with a form in two variables.
In the course of Christoffel’s work there arise certain functions (ikrs); these were originally found by Riemann in 1861 in his investigations on the curvature of hypersurfaces. For a surface in space they reduce to the one quantity K
7. Ricci and Levi-Civita.
To Christoffel is due a method whereby from invariants involving derivatives of the fundamental form and of the functions ϕ may be derived invariants involving higher derivatives. This process has been called by Ricci and Levi-Civita covariant derivation, and they have made it the base of their researches in this subject. These researches have been collected and given by them in complete form in the Mathematische Annalen||, and on their work they have based a calculus which they call Absolute differential calculus. They give a complete solution of the problem, and show that in order to determine all differential invariants of order μ, it is sufficient to determine the algebraic invariants of the system:
(1) The fundamental differential quantic,
(2) The covariant derivatives of the arbitrary functions ϕ up to the order μ,
(3) A certain quadrilinear form G4 and its covariant derivatives up to the order μ – 2*.
8. Lie.
The second method is founded on the theory of groups of Lie, and is a direct application of the theory given in his paper Ueber Differentialinvarianten†. This theory involves the use of infinitesimal transformations, and the invariants are obtained a...
