Geometrodynamics based on geodesic equation with Cartan structural equation on Riemannian manifolds

Abstract

Motivated by the geospin matrix $W$ as a new variable in Riemannian geometry. Then, we use four real dynamical variables $\left\{ a,\alpha ,v,W \right\}$ to show the dynamical essence of Cartan structural equation, we obtain the geometrodynamics on Riemannian manifolds that can be expressed below \begin{align} & \Theta /d{{t}^{2}}=a-v\wedge W,\nonumber & d\Theta /d{{t}^{3}}=v\wedge \alpha -a\wedge W,\nonumber & \Omega /d{{t}^{2}}=\alpha -W\wedge W, \nonumber & d\Omega /d{{t}^{3}}=W\wedge \alpha -\alpha \wedge W\nonumber \end{align} that is valid more generally for any connection in a principal bundle. We can see that the first equation explains the geodesic equation very well, the second formula means the first Bianchi identity, while the last equation reveals the dynamical nature of the second Bianchi identity. It implies that Cartan structural equations on Riemannian manifolds can be rewritten in a real geometrodynamical form based on the geospin matrix.