  For a surface in 3-sphere, by identifying the conformal round 3-sphere as the
projectivized positive light cone in Minkowski 5-spacetime, we use the
conformal Gauss map and the conformal transform to construct the associate
homogeneous 4-surface in Minkowski 5-spacetime. We then derive the local
fundamental theorem for a surface in conformal round 3-sphere from that of the
associate 4-surface in Minkowski 5-spacetime. More importantly, following the
idea of Fefferman and Graham, we construct local scalar invariants for a
surface in conformal round 3-sphere. One distinct feature of our construction
is to link the classic work of Blaschke to the works of Bryan and
Fefferman-Graham.
