integrais DE ANCELMO L. GRACELI.

Ricci curvature tensor

It is a standard exercise of (multi)linear algebra to verify that this definition does not depend on the choice of the basis ⁠${\displaystyle {�}_1,\dots ,{�}_{�}}$⁠.

In abstract index notation,$${\displaystyle {\mathrm{R}\mathrm{i}\mathrm{c}}_{��}={\mathrm{R}}^{�}{}_{���}={\mathrm{R}}^{�}{}_{���}.}$$

Sign conventions. Note that some sources define ${\displaystyle �(�,�)�}$ to be what would here be called ⁠${\displaystyle -�(�,�)�}$⁠; they would then define ${\displaystyle {\mathrm{Ric}}_{�}}$ as ⁠${\displaystyle -\mathrm{tr}(�\mapsto {\mathrm{R}}_{�}(�,�)�)}$⁠. Although sign conventions differ about the Riemann tensor, they do not differ about the Ricci tensor.

INTEGRAIS DE ANCELMO L. GRACELI 

                                       

X [ ]  =  -S  t []   dt

                                                             

X [] = Z  t []  []  []  dt

X []  [ ]= []Z  t []    []  dt

integrais DE ANCELMO L. GRACELI.

Ricci curvature tensor

It is a standard exercise of (multi)linear algebra to verify that this definition does not depend on the choice of the basis ⁠⁠.

In abstract index notation,

Sign conventions. Note that some sources define  to be what would here be called ⁠⁠; they would then define  as ⁠⁠. Although sign conventions differ about the Riemann tensor, they do not differ about the Ricci tensor.

INTEGRAIS DE ANCELMO L. GRACELI 

                                       

X [ ]  =  -S  t [] =  dt

                                                             

X [] = Z  t []=   []  []  dt

X []  [ ]= []Z  t [] =   []  dt