Complete Contraction of Tensor and Its Application

  •  Jinxiong Xing    
  •  Yifan Gong    


This article presents a mathematical model for tensor complete contraction utilizing permutation group theory. The model helps in identifying the index class that dictates the complete contraction of even-order tensors. Additionally, the concept of a complete contraction closed loop is introduced along with a method for calculating the number of completely contracted images of any even-order tensor under any λ type permutation. The model is applied to investigate the algebraic structure of global conformal invariants on three-dimensional hypersurfaces. Moreover, it is shown that the generalized Willmore functional is the sole global conformal invariant in three-dimensional hypersurfaces when the difference is a constant multiple.

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