Complex velocities on real manifolds
Ivan Kolář · Czechoslovak Mathematical Journal · 1971
The paper starts with the definitions of complex velocities and of complex contact elements of order r on a real manifold of class C.These concepts are introduced inevitably in a formal way, nevertheless, they represent a theoretical basis for the use of imaginary elements in many problems in differential geometry.Our main conclu sion is that one may consider some kinds of imaginary elements up to order г on every real manifold of class C. To give a concrete illustration of our ideas, we discuss the asymptotic directions at an elliptic point of a surface of class C^ in the real pro jective 3-dimensional space.In the remaining part of the paper, the general concept of a complex r-jet of a real manifold of class С into a real manifold of the same class is introduced and the complex prolongations of real fibered manifolds are treated.