Unparametrized

For many metrics the tangent space T#95;c#92;operatorname#123;Imm#125;(S¹,#92;mathbbR²) at each curve c splits into vertical and horizontal components (with respect to the projection onto the shape space B#95;i(S¹,#92;mathbbR²)#61;#92;operatorname#123;Imm#125;(S¹,#92;mathbbR²)#47;#92;operatorname#123;Diff#125;(S¹) of unparametrized curves and with respect to the metric G).