Induced connections of two types on a surface of an affine space

In the affine space the fundamental-group connection in the bundle associated with a surface as a manifold of tangent planes is investigated. The principal bundle contains a quotient bundle of tangent frames, the typical fiber of which is a linear group acting in a centered tangent plane and a quotient bundle of normal frames, the typical fiber of which is a linear group acting inefficiently in a normal quotient space. The curvature object of the fundamental-group connection is a tensor that contains two primary subtensors tangent and normal linear connections. The tensor of non-absolute parallel transference is constructed. Two envelopment of the connection object is obtained. Analytic and geometric conditions of coincidence of two types of envelopment are found. The covariant derivatives of equipping quasitensor form a tensor. The alternations of the covariant derivatives of the objects of the affine and linear connections of the first type are equal to the corresponding components of the curvature tensor and for the second type they vanish.