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Geometry of conformal mechanics

Conformal mechanics, the simplest conformal field theory, is reformulated as a d=1 nonlinear sigma model on the group SO(1,2). Its action and equation of motion are shown to have a simple representation in terms of corresponding Cartan forms. The equation of motion amounts to certain algebraic relations between these forms which define a class of geodesics on SO(1,2). The authors' geometric approach demonstrates deep analogies between the d=1 conformal mechanics and the d=2 Liouville theory. It is equally applicable to more complicated cases of superconformal mechanics and can be used to deduce the equations of the latter in a manifestly invariant superfield form.