We show that a compact Riemannian
3
3
-manifold
M
M
with strictly convex simply connected boundary and sectional curvature
K
≤
a
≤
0
K\leq a\leq 0
is isometric to a convex domain in a complete simply connected space of constant curvature
a
a
, provided that
K
≡
a
K\equiv a
on planes tangent to the boundary of
M
M
. This yields a characterization of strictly convex surfaces with minimal total curvature in Cartan-Hadamard
3
3
-manifolds, and extends some rigidity results of Greene-Wu, Gromov, and Schroeder-Strake. Our proof is based on a recent comparison formula for total curvature of Riemannian hypersurfaces, which also yields some dual results for
K
≥
a
≥
0
K\geq a\geq 0
.