Mth 497C Sep 17, 004 1 Curves nd Surfces Fll 004, PSU Lecture Notes 3 1.8 The generl defnton of curvture; Fox-Mlnor s Theorem Let α: [, b] R n be curve nd P = {t 0,...,t n } be prtton of [, b], then the pproxmton of the totl curvture of α wth respect to P s defned s n 1 totl κ[α, P ]:= ngle αt αt 1,αt +1 αt, =1 nd the totl curvture of α s gven by totl κ[α] := sup{ κ[α, P ] P P rtton[, b] }. Our mn m here s to prove the followng observton due to Rlph Fox nd John Mlnor: Theorem 1 Fox-Mnor. If α: [, b] R n s C unt speed curve, then totl κ[α] = α t dt. Ths theorem mples, by the men vlue theorem for ntegrls, tht for ny t, b, 1 [ κt = lm ɛ 0 ɛ totl κ α ] t+ɛ. t ɛ The bove formul my be tken s the defnton of curvture for generl not necessrly C curves. To prove the bove theorem frst we need to develop some bsc sphercl geometry. Let 1 Lst revsed: September 17, 004 S n := {p R n+1 p =1}. 1
denote the n-dmensonl unt sphere n R n+1. Defne mppng from S n S n to R by dst S np, q := nglep, q. Exercse. Show tht S n, dst S n s metrc spce. The bove metrc hs smple geometrc nterpretton descrbed s follows. By gret crcle C S n we men the ntersecton of S n wth two dmensonl plne whch psses through the orgn o of R n+1. For ny pr of ponts p, q S, there exsts plne pssng through them nd the orgn. When p ±q ths plne s gven by the lner combntons of p nd q nd thus s unque; otherwse, p, q nd o le on lne nd there exsts nfntely mny two dmensonl plnes pssng through them. Thus through every prs of ponts of S n there psses gret crcle, whch s unque whenever p ±q. Exercse 3. For ny prs of ponts p, q S n, let C be gret crcle pssng through them. If p q, let l 1 nd l denote the length of the two segments n C determned by p nd q, then dst S np, q = mn{l 1,l }.Hnt: Let p C be vector orthogonl to p, then C my be prmetrzed s the set of ponts trced by the curve p cost+p snt. Let α: [, b] S n be sphercl curve,.e., Euclden curve α: [, b] R n+1 wth α = 1. For ny prtton P = {t 0,...,t n } of [, b], the sphercl length of α wth respect the prtton P s defned s length S n[α, P ]= n dst S n αt,αt 1. =1 The norm of ny prtton P of [, b] s defned s P := mx{t t 1 1 n}. If P 1 nd P re prtons of [, b], we sy tht P s refnement of P 1 provded tht P 1 P. Exercse 4. Show tht f P s refnement of P 1, then length S n[α, P ] length S n[α, P 1 ]. Hnt:Use the fct tht dst S n stsfes the trngle nequlty, see Exc..
The sphercl length of α s defned by length S n[α] = sup { length S n[α, P ] P P rtton[, b] }. Lemm 5. If α: [, b] S n s unt speed sphercl curve, then length S n[α] = length[α]. Proof. Let P k := {t k 0,...,t k n} be sequence of prttons of [, b] wth lm P k =0, nd θ k := dst S n α k t,α k t 1 = ngle α k t,α k t 1 be the correspondng sphercl dstnces. Then, snce α hs unt speed, θ k sn = αt k αt k 1 t k t k 1 P k. In prtculr, θ k lm sn =0. Now, snce lm x 0 snx/x = 1, t follows tht, for ny ɛ>0, there exsts N>0, dependng only on P k, such tht f k>n, then θ 1 ɛθ k k sn 1 + ɛθ k, whch yelds tht 1 ɛ length S n[α, P k ] length[α, P k ] 1 + ɛ length S n[α, P k ]. The bove nequltes re stsfed by ny ɛ>0provded tht k s lrge enough. Thus lm length S n[α, P k ] = length[α]. Further, note tht f P s ny prttons of [, b] we my construct sequnce of prttons by successve refnements of P so tht lm P k =0. By Exercse 4, length S n[α, P k ] length S n[α, P k+1 ]. Thus the bove expresson shows tht, for ny prtton P of [, b], length S n[α, P ] length[α]. 3
The lst two expressons now yed tht sup{ length S n[α, P ] P P rtton[, b] } = length[α], whch completes the proof. Exercse 6. Show tht f P s refnement of P 1, then totlκ[α, P ] totlκ[α, P 1 ]. Now we re redy to prove the theorem of Fox-Mlnor: Proof of Theorem 1. As n the proof of the prevous lemm, let P k = {t k 0,...,t k n} be sequence of prttons of [, b] wth lm P k = 0. Set θ k := ngle αt k αt k 1,αt k +1 αt k, where =1,...,n 1. Further, set nd Recll tht, by the prevous lemm, lm t k := tk + t k 1 φ k := ngle α t k,α t k +1. φ k = length S n 1[α ] = length[α ]= α t dt. Thus to complete the proof t suffces to show tht, for every ɛ>0, there exsts N such tht for ll k N, for then t would follow tht θ k φ k ɛt k +1 t k 1; 1 ɛ[, b] θ k φ k ɛ[, b], whch would n turn yeld lm totl κ[α, P k ] = lm θ k = lm 4 φ k = α t dt.
Now, smlr to the proof of Lemm 5, note tht gven ny prtton P of [, b], we my construct by subsequent refnements sequnce of prttons P k, wth P 0 = P, such tht lm P k = 0. Thus the lst expresson, together wth Excercse 6, yelds tht totlκ[α, P ] α t dt. The lst two expressons complete the proof; so t remns to estblsh 1. To ths end let := ngle α t k, αt k αt k 1. β k By the trngle nequlty for ngles Exercse. whch yelds φ k β k + θ k + β k +1, nd θ k β k + φ k + β k +1, φ k θ k β k + β k +1. So to prove 1 t s enough to show tht for every ɛ>0 β k ɛ t t 1 provded tht k s lrge enough. See Exercse 7. Exercse* 7. Let α: [, b] R n be C curve. For every t, s [, b], t s, defne t + s ft, s := ngle α,αt αs. Show tht ft, s lm t s t s =0. In prtculr, f we set ft, t = 0, then the resultng functon f :[, b] [, b] R s contnuous. So, snce [, b] s compct, f s unformly contnuous,.e., for every ɛ>0, there s δ such tht ft fs ɛ, whever t s δ. Does ths result hold for C 1 curves s well? 5