The Lie group, SE(3), made up of all rigid-body motions is called the special Euclidean group in three dimensions, on which mathematical notions are set up for robotic manipulation or trajectory planning. This thesis deals with a systematic study of metrics and connections on SE(3). Three connections, called (-)-,(+)- and (0)-connections after Cartan and Schouten, are introduced, and further two metric connections are defined through two types of left-invariant metrics. The Cartesian stiffness matrices are computed with respect to the connections introduced, which will play a significant role in the stability analysisn for multifingered grasp.