The submanifolds $X_m$ of the manifold $^*g-MEX_n$. II. Fundamental equations on $X_m$ of $^*g-MEX_n$

Abstract

In our previous paper [4], we studied the induced connection of the $^*g-Me$-connection on a submanifold $X_m$ embedded in a manifold $^*g-Mex_n$ together with the generalized coefficients $ω_{ij}$ of the second fundamental form of $X_m$, with emphasis on the proof of a necessary and sufficient condition for the induced connection of $X_m$ in $^*g-MEX_n$ to be a $^*g-ME$-connection. This paper is a direct continuation of [4]. In this paper, we derive the generalized fundamental equations on $X_m$ of $^*g-MEX_n$, such as the generalized Gauss formulae, the generalized Weingarten equations, and the Gauss-Codazzi equations. Furthermore, we also present surveyable tensorial representations of curvature tensors $R^\mu_{ω\muλ}$ of $^*g-MEX_n$ and $R^h_{ijk}$ of $X_m$.