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Coordinates play an essential role in the description of real world objects and physical processes. In robotics, coordinates are used to describe the kinematic structure and the kinematic and dynamic behavior. The description mostly takes place in charts, assigned by the observer of the robotic system. However, it is crucial that the described physical process does not depend on the coordinate choice of the observer. In this work we show the relation between coordinates and manipulability analysis. Manipulability measures are dependent of joint coordinates of the robot and task coordinates in the workspace of the robot. Both relations can be analyzed with tensor geometry. We remove the dependency on joint coordinates through the use of an appropriate metric. With the help of tensor contraction, the resulting induced metric in the workspace can be transformed into a coordinate invariant matrix. After applying eigenvalue decomposition on this matrix, we can visualize the dynamic manipulability of a robot as a coordinate invariant ellipsoid.