  Two types of finite element spaces on triangles are constructed for div-div
conforming symmetric tensors. Besides the normal-normal continuity, the stress
tensor is continuous at vertices and another trace involving combination of
derivatives of stress is identified. Polynomial complex, finite element
complex, and Hilbert complex are presented and a commuting diagram between them
is given. The constructed div-div conforming elements are exploited to
discretize the mixed formulation of the biharmonic equation. Optimal order and
superconvergence error analysis is provided. By rotation, finite elements for
rot-rot conforming symmetric strain are also obtained.
