Could someone tell me why $$\overline{\operatorname{Ric}}(U,V) = \operatorname{Ric}_M(U^* ,V^*)+ (n((\log f)')^2+(\log f)'')\langle U,V \rangle -(n-1)(\log f)'' \langle U, \partial_t \rangle \langle V, \partial_t \rangle $$
where $U$ and $V$ are arbitrary fields of $-I \times_f M$ e $U^* = (\pi_M)_*(U)$ denotes the projection on fiber M of a vector field U defined in $-I \times _f M^n$, that is,Aqui, $\operatorname{Ric}_M$denotes the Ricci tensor of the Riemannian variety $M^n$ e $\overline{\operatorname{Ric}}(U,V)$ denotes the Ricci tensor of the Riemannian variety $I \times_f M$? the article asks to use corollary 7.43 from O'neill's book the article asks to use corollary 7.43 from O'neill's book. $$\overline{\operatorname{Ric}}(U,V) = \operatorname{Ric}_M(U^* ,V^*)+ (n((\log f)')^2+(\log f)'') \langle U,V \rangle $$ I couldn't make the other part appear.