In this paper, we present a nonlinear version of the linear elasticity (Calabi, Kr\"oner, Riemannian deformation) complex which encodes isometric embedding, metric, curvature and the Bianchi identity. We reformulate the rigidity theorem and a fundamental theorem of Riemannian geometry as the exactness of this complex. Then we generalize an algebraic approach for constructing finite elements for the Bernstein-Gelfand-Gelfand (BGG) complexes. In particular, we discuss the reduction of degrees of freedom with injective connecting maps in the BGG diagrams. We derive a strain complex in two space dimensions with a diagram chase.

翻译：本文提出线性弹性（Calabi、Krőner、黎曼形变）复形的非线性版本，该复形涵盖了等距嵌入、度量、曲率以及比安基恒等式。我们将刚性定理和黎曼几何基本定理重新表述为该复形的正合性。随后，我们推广了一种用于伯恩斯坦-盖尔范德-盖尔范德（BGG）复形的有限元构造代数方法。特别地，我们讨论了在BGG图表中通过单射连接映射减少自由度的问题。通过图表追踪，我们推导出二维空间中的应变复形。