Order of Holonomy of a Surface with Projective Connection

Abstract

A submanifold in a space with Cartan connection, see [3], represents a natural generalization of a submanifold in the corresponding homogeneous space. E. Cartan himself showed in the case of a surface in a 3-dimensional space with projective connection, [1], that his method of specialization of frames can also be applied to the investigation of these submanifolds. A. Svec pointed out, cf. [5], that such a submanifold can be considered as a separate structure. From this point of view, a surface in a 3-space with projective connection is called a manifold of type P 0,3 2 , or shortly, a surface with projective connection. Naturally, differential geometry of a surface p with projective connection differs from differential geometry of a surface in projective 3-space P 3 . In this paper, we want to show that the difference between p and a surface in P 3 can be also measured in individual orders. If we use the computational procedures by E. Cartan, then the difference in order k between p and a surface in P 3 is characterized by the difference between the formulae of the (k - 1)-st prolongation for p and the formulae of the (k - 1)-st prolongation for a surface in P 3. Conversely, if these formulae coincide, then we say that p is holonomic of order k, or, shortly, k-holonomic. Dealing with the first prolongation, we show the invariance of the condition for 2-holonomy also in a formal computional way, but we do not repeat it for higher orders, since we present a direct invariant definition of k-holonomy for an arbitrary manifold with connection in [4].