Con-circular Transformations On Differential Manifolds

We define a semi-Riemannian space of index 1.It is the natural extension of Minkowski space.The semi-Riemann space and the Riemann space have obvious differences,The metric matrix of semi-Riemann space is non-degenerate matrix,The metric matrix of Riemann space is positive matrix.Therefore,the inner product of two non-zero vectors in the tangent space of any point in the semi-Riemann space can be greater than zero,equal to zero or less than zero.Because the inner product is more complicated,the classification of curves becomes more complicated.In a remi-Riemann space,we have timelike,spacelike,and lightlike curves,timelike and spacelike plane constant curvature curves are defined as timelike geodesic circle and space like geodesic circle.Then we define timelike and spacelike geodesic circle and discuss the necessary and sufficient condition of a conformal transformation transform every timelike geodesic circle and spacelike geodesic circle into timelike geodesic circle and spacelike geodesic circle.