  Flat sheets encoded with patterns of contraction/elongation morph into curved
surfaces. If the surfaces bear Gauss curvature, the resulting actuation can be
strong and powerful. We deploy the Gauss-Bonnet theorem to deduce the Gauss
curvature encoded in a pattern of uniform-magnitude contraction/elongation with
spatially varying direction, as is commonly implemented in patterned liquid
crystal elastomers. This approach reveals two fundamentally distinct
contributions: a structural curvature which depends on the precise form of the
pattern, and a topological curvature generated by defects in the contractile
direction. These curvatures grow as different functions the
contraction/elongation magnitude, explaining the apparent contradiction between
previous calculations for simple +1 defects, and smooth defect-free patterns.
We verify these structural and topological contributions by conducting
numerical shell calculations on sheets encoded with simple higher-order
contractile defects to reveal their activated morphology. Finally we calculate
the Gauss curvature generated by patterns with spatially varying magnitude and
direction, which leads to additional magnitude gradient contributions to the
structural term. We anticipate this form will be useful whenever magnitude and
direction are natural variables, including in describing the contraction of a
muscle along its patterned fiber direction, or a tissue growing by elongating
its cells.
