Induction

Abstract

If we define, in ordinary euclidean three-space R ₃, a manifold X ₂ by the equations x ^x = x ^x(u, v), x =1, 2, 3, then a measurement is determined in this X ₂:

$$ d{s^2} = Ed{u^2} + 2Fdudv + Gd{v^2},E = {\sum\limits_v {\left( {\frac{{\partial {x^v}}}{{\partial u}}} \right)} ^2}, $$

and this linear element defines a Riemannian connection V ₂ in the X ₂. We say that the Riemannian connection is induced into the X ₂ by the euclidean connection of the R ₃. When Levi-Civita, in 1917, demonstrated the possibility of a parallel displacement in a V ₂, he did it by just such a process of induction⁷. Since that time the method has often been used to obtain a differential geometry of X _m imbedded in an X _n with a certain connection. Riemannian geometry in a V _n leads to a Riemannian geometry in an imbedded X _m , a plane affine geometry in an E _n to an A _m in an imbedded X _m .⁸ The “generalized absolute calculus” of Vitali is founded upon this principle⁹. We shall first show how it can be applied to an L _n .