  The two dimensional version of the Sen connection for spinors and tensors on
spacelike 2-surfaces is constructed. A complex metric $\gamma_{AB}$ on the spin
spaces is found which characterizes both the algebraic and extrinsic
geometrical properties of the 2-surface $\$ $. The curvature of the two
dimensional Sen operator $\Delta_e$ is the pull back to $\$ $ of the
anti-self-dual part of the spacetime curvature while its `torsion' is a boost
gauge invariant expression of the extrinsic curvatures of $\$ $. The difference
of the 2 dimensional Sen and the induced spin connections is the anti-self-dual
part of the `torsion'. The irreducible parts of $\Delta_e$ are shown to be the
familiar 2-surface twistor and the Weyl--Sen--Witten operators. Two Sen--Witten
type identities are derived, the first is an identity between the 2 dimensional
twistor and the Weyl--Sen--Witten operators and the integrand of Penrose's
charge integral, while the second contains the `torsion' as well. For spinor
fields satisfying the 2-surface twistor equation the first reduces to Tod's
formula for the kinematical twistor.
