we discuss curved target spaces and higher p – brane theories such as the (super –)membrane [@10] which, in contrast to string theory, is an interacting theory even on flat target spaces. These issues will be discussed in our companion paper [@11] which overlaps in part with the pioneering work [@11a] but, as the present paper, departs from it in most aspects. See also [@11b] for a different new approach to string theory using a modification of the Lorentz group.

The Nambu – Goto String {#s2}
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In introductory texts to string theory [@1] the Nambu – Goto string is barely mentioned, one almost immediately switches to the Polyakov string whose corresponding action has the same set of classical solutions as the Nambu – Goto action. The advantage of the Polyakov action is that it is bilinear in the string degrees of freedom on flat target space, however, this comes at the price of introducing an additional Weyl invariance and an auxiliary worldsheet metric. From the point of a geometer this classical reformulation of the geometerical object, the string, as a theory of scalars interacting with topological gravity is rather unnatural. We thus review below the canonical formulation of the 