▁and ▁South ▁American ▁football , ▁from ▁ 1 9 7 1 ▁( year ▁of ▁Brasile ir ão ' s ▁creation ) ▁to ▁the ▁most ▁recent ▁completed ▁season . ▁ ▁Se asons ▁ ▁Brasile ir ão ▁era ▁ ▁References ▁ ▁External ▁links ▁ ▁Se asons ▁Portug ues a <0x0A> </s> ▁In ▁differential ▁geometry , ▁Sant al ó ' s ▁formula ▁describes ▁how ▁to ▁integrate ▁a ▁function ▁on ▁the ▁unit ▁sphere ▁bundle ▁of ▁a ▁Riemann ian ▁manifold ▁by ▁first ▁integr ating ▁along ▁every ▁ge odes ic ▁separately ▁and ▁then ▁ ▁over ▁the ▁space ▁of ▁all ▁ge odes ics . ▁It ▁is ▁a ▁standard ▁tool ▁in ▁integral ▁geometry ▁and ▁has ▁applications ▁in ▁is oper imet ric ▁and ▁rig id ity ▁results . ▁The ▁formula ▁is ▁named ▁after ▁Luis ▁Sant al ó , ▁who ▁first ▁proved ▁the ▁result ▁in ▁ 1 9 5 2 . ▁ ▁Form ulation ▁▁ ▁Let ▁ ▁be ▁a ▁compact , ▁orient ed ▁Riemann ian ▁manifold ▁with ▁boundary . ▁Then ▁for ▁a ▁function ▁, ▁Sant al ó ' s ▁formula ▁takes ▁the ▁form ▁ ▁where ▁▁▁ ▁is ▁the ▁ge odes ic ▁flow ▁and ▁ ▁is ▁the ▁exit ▁time ▁of ▁the ▁ge odes ic ▁with ▁initial ▁conditions ▁, ▁▁ ▁and ▁ ▁are ▁the ▁Riemann ian ▁volume ▁forms ▁with ▁respect ▁to ▁the ▁S as aki ▁metric ▁on ▁ ▁and ▁ ▁respectively ▁( ▁is ▁also ▁called ▁Li ou ville ▁measure ), ▁▁ ▁is ▁the ▁in ward - point ing ▁unit ▁normal ▁to ▁ ▁and ▁ ▁the ▁infl ux - bound ary , ▁which ▁should ▁be ▁thought