Harmonic maps of Lorentz surfaces, quadratic differentials and paraholomorphicity (Q1356593)

In \(\mathbb{R}^2\), the author identifies \((x,y)\) with \(x+\varepsilon y\), where \(\varepsilon \) is an indeterminate with \(\varepsilon ^2=1\) and obtains a commutative, associative and unitary algebra \(\mathbb{A}\) over \(\mathbb{R}\) of rank \(2\) with zero divisors. \(\mathbb{A}\) is called the algebra of paracomplex numbers or double numbers. In this paper some paracomplex analysis and parageometry is developed. Some relations are established between harmonic maps \(f\) of Lorentz surfaces \(M\) into indefinite Riemannian manifolds and paracomplex analysis including a theory of quadratic and higher order differentials associated to \(f\) as in the Riemannian case.