▁follows ▁from ▁the ▁following ▁two ▁ing red ients , ▁not ing ▁that ▁ ▁has ▁measure ▁zero . ▁ ▁An ▁integration ▁by ▁parts ▁formula ▁for ▁the ▁ge odes ic ▁vector ▁field ▁: ▁▁ ▁The ▁construction ▁of ▁a ▁resol vent ▁for ▁the ▁transport ▁equation ▁: ▁ ▁For ▁the ▁integration ▁by ▁parts ▁formula , ▁recall ▁that ▁ ▁leaves ▁the ▁Li ou ville - measure ▁ ▁invariant ▁and ▁hence ▁, ▁the ▁diver gence ▁with ▁respect ▁to ▁the ▁S as aki - metric ▁. ▁The ▁result ▁thus ▁follows ▁from ▁the ▁diver gence ▁theorem ▁and ▁the ▁observation ▁that ▁, ▁where ▁ ▁is ▁the ▁in ward - point ing ▁unit - normal ▁to ▁. ▁The ▁resol vent ▁is ▁explicitly ▁given ▁by ▁ ▁and ▁the ▁ ▁mapping ▁property ▁ ▁follows ▁from ▁the ▁smooth ness ▁of ▁, ▁which ▁is ▁a ▁consequence ▁of ▁the ▁non - tra pping ▁and ▁the ▁convex ity ▁assumption . ▁ ▁References ▁▁ ▁Category : D iffer ential ▁geometry ▁Category : Math emat ical ▁concepts ▁Category : Integr als ▁Category : R iemann ian ▁geometry <0x0A> </s> ▁I ▁Got ▁Ne eds ▁may ▁refer ▁to : ▁" I ▁Got ▁Ne eds ", ▁a ▁ 1 9 9 7 ▁television ▁episode ▁in ▁the ▁first ▁season ▁of ▁B ump ▁in ▁the ▁Night ▁" I ▁Got ▁Ne eds ", ▁a ▁ 1 9 9 5 ▁song ▁on ▁American ▁ra pper ▁Count ▁Bass ▁D ' s ▁album ▁Pre - Life ▁Cris is <0x0A> </s> ▁The ▁M 1 ▁Gar and ▁is ▁a ▁. 3 0 - 0 6 ▁cal iber ▁semi - autom atic ▁rif