This paper presents a geometrically and dynamically transparent projected predictive Energy-Bounding Approach (EBA) for multiple degree-of-freedom (m-DOF) haptic teleoperation with time delays. The straightforward extension of a single-DOF teleoperation to m-DOF teleoperation suffers from a bad force distortion problem that becomes worse with the increase in communication time delays between master and slave sites. Due to this, geometry perception of remote m-DOF objects at master site is severely deteriorated. To solve this problem, a projected predictive m-DOF EBA is proposed by combining the m-DOF Predictive EBA and the projection method that can compensate the force distortion problem. The proposed approach also includes steps to get stably the information about the contact locations (geometry) of m-DOF remote objects without geometry detection sensors. In order to validate the proposed approach, some experiments performing surface contour-following are conducted using two Phantoms: (i) 2-DOF object (bobbin), and (ii) 3-DOF semi-spherical object in the presence of large time delays.