  This work proposes a natural extension of the Bargmann-Fock representation to
a SUSY system. The main objective is to show that all essential structures of
the n-dimensional SUSY oscillator are supplied by basic differential
geometrical notions on an analytical R^n, except for the scalar product which
is the only additional ingredient. The restriction to real numbers implies only
a minor loss of structure but makes the essential features clearer. In
particular, euclidean evolution is enforced naturally by identification with
the 1-parametric group of dilations.
