The differential geometry of an imbedded (e.g. string or membrane world sheet) surface in a higher-dimensional background is shown to be conveniently describable (except in the null limit case) in terms of what are designated as its first, second, and third fundamental tensors, which will have the respective symmetry properties η μν = η ( μν) as a trivial algebraic identity, K μν ρ = K ( μν) ρ as the “generalised Weingarten identity”, which is the (Frobenius type) integrability condition for the imbedding, and Ξ λμν ρ = Ξ ( λμν) ρ as a “generalised Codazzi equation”, which depends on the background geometry being flat or of constant curvature, needing replacement by a more complicated expression for a generic value of the background curvature B κλ μ ν . The “generalised Gauss equation” expressing the dependence on this background curvature of the internal curvature tensor R κλ μ ν of the imbedded surface is converted into terms of the first and second fundamental tensors, and it is thereby demonstrated that the vanishing of the (conformally invariant) “conformation tensor”, i.e. the trace free part C μν ρ of the second fundamental tensor K μv ρ , is a sufficient condition for conformal flatness of the imbedded surface (and thus in particular for the vanishing of its (Weyl type) conformal curvature tensor C κλ μ ν ) provided the background is itself conformally flat. In a trio of which the first two members are the generalised Gauss and Codazzi equations, the “third” member is shown to give an expression in terms of C μν ρ for the (trace free, conformally invariant) “outer curvature” tensor Ω κλ μ ν whose vanishing is the condition for feasibility of the natural generalisation of the Walker frame transportation ansatz. The vanishing of C μν ρ is shown to be sufficient in a conformally flat background for the vanishing also of Ω κλ μ ν .