BASIC DIFFERENTIAL GEOMETRY: VARIATIONAL THEORY OF GEODESICS

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1 BASIC DIFFERETIAL GEOMETRY: VARIATIOAL THEORY OF GEODESICS WERER BALLMA Introduction We ssume throughout tht M is Riemnnin mnifold. For points p, q in M, we sk for the criticl points of the length functionl L on the spce Ω ps pq of piecewise smooth curves from p to q. Since Ω ps pq is not mnifold nd L is not smooth, the notion of criticl point nd derivtive of L require n explntion. Following one of the possible definitions of tngent vector t point in smooth mnifold, we consider smooth curves s c s Ω ps pq, ε < s < ε, of piecewise smooth curves in Ω ps pq so tht the length L(s) := L(c s ) depends smoothly on s. Then s L(0) represents the derivtive of L t c = c 0 in the direction represented by the curve s c s in Ω ps pq. The notion of smooth curves in Ω ps pq is mde precise by the concept of proper piecewise smooth vrition nd the corresponding derivtive s L(0) is obtined in the First Vrition Formul. As n ppliction of the First Vrition Formul we obtin tht the set of criticl points of the length functionl L on Ω ps pq consists precisely of the geodesic segments from p tp q. For smooth function f on smooth mnifold, the second derivtive of f is well-defined in the criticl points of f. 1 Correspondingly we cn define the second derivtive of the length functionl for geodesic segment c in Ω ps pq. The Second Vrition Formul determines this second derivtive in terms of the geometry of M long c. Lst updte: In generl, there is no resonble wy of defining the second derivtive of f t point, except the point is criticl. 1

2 2 Bsic Differentil Geometry Contents Introduction 1 Conventions 2 1. First nd Second Vrition of Arc Length First Vrition of Arc Length Geodesic Segments s Criticl Points Second Vrition of Arc Length 6 2. Boundry Conditions 7 3. Energy Versus Arc Length Semi-Riemnnin metrics 10 Acknowledgment 10 Conventions The covrint derivtive of vector field X long piecewise smooth curve c is denoted by X. Correspondingly, the covrint derivtive of c is denoted c.

3 Vritionl Theory of Geodesics 3 1. First nd Second Vrition of Arc Length Let c : [, b] M be piecewise smooth curve. By definition, the length L(c) of c is (1.1) L(c) = c (t) dt. The length of c is invrint under reprmeteriztion. More generlly, if ϕ : [α, β] [, b] is piecewise smooth, (wekly) monotone nd surjective mp, then L(c ϕ) = L(c). A piecewise smooth n-prmeter vrition of c is mp H : U [, b] M, where U is n open neighborhood of 0 in R n, such tht H(0, ) = c nd such tht there is subdivision = t 0 <... < t k = b of [, b] such tht H is smooth on the sets U [t i 1, t i ], 1 i k. We sy tht vrition H of c is proper if H(s, ) = c() nd H(s, b) = c(b) for ll s U. We consider vritions s formliztion of the ide of smooth fmilies of curves. For given vrition H of c, we set c s = H(s, ). If H is 1-prmeter vrition, then U is n open intervl nd the fmily (c s ) is smooth curve in the bove sense in the spce of piecewise smooth curves. We my then consider the vrition field V = s H(0, ) s the tngent vector to this curve t time s = 0. If H is proper, V () = 0 nd V (b) = 0. ote tht V is piecewise smooth. Lemm 1.1. Let c : [, b] M be piecewise smooth nd V be piecewise smooth vector field long c. Then there is piecewise smooth 1-prmeter vrition H of c with vrition field V nd with H(s, t) = c(t) for ll (s, t) with V (t) = 0. In prticulr, H is proper if V () = 0 nd V (b) = 0. Remrk 1.2. The vrition H in the Lemm is not unique. This is similr to the fct tht tngent vector in mnifold is not represented by unique smooth curve through the foot point of the vector but rther by n equivlence clss of such curves. Proof of Lemm 1.1. Set H(s, t) := exp(s V (t)) First Vrition of Arc Length. Since the length of curve is invrint under reprmeteriztion, we let c : [, b] M be piecewise smooth curve with constnt speed v 0, tht is, c (t) = v for ll t [, b]. First Vrition Formul 1.3. Let H : U [, b] M be piecewise smooth 1-prmeter vrition of c. Let V be the vrition field of H nd set c s := H(s, ), L(s) := L(c s ). Then L : U R is smooth bout 0 nd s L(0) = 1 { v V, c b k 1 + V (t i ), c (t i ) } V, c dt,

4 4 Bsic Differentil Geometry where the subdivision = t 0 <... < t k = b of [, b] is chosen such tht c is smooth on the sets [t i 1, t i ], nd c (t i ) := c (t i 0) c (t i + 0). Proof. By ssumption we hve c (t) 0 for ll t [, b]. Hence by diminishing the size of U if necessry, we cn ssume c s(t) 0 for ll s U nd t [, b]. Then c s(t) is smooth on the sets U [t i 1, t i ] nd hence L is smooth in s. Moreover, s L(0) = s { k { k = = ti ti s t i 1 ti k = 1 v = 1 v t i 1 c s(t) dt} s=0 t i 1 ( ) } c s(t) dt s=0 D s t H(0, t), t H(0, t) dt t i 1 c 0(t) k ti D t s H(0, t), c (t) dt t i 1 k ti { } t V, c V, D t c dt = 1 { k v V, c t i b t i 1 } V, D t c dt. We cll s L(0) the first vrition of rc length with respect to the given vrition. The first vrition of rc length corresponds to the derivtive of the length functionl in the direction of the tngent vector V of the vrition. In fct, s L(0) only depends on V, not on the prticulr choice of vrition H defining V. This is in ccordnce with derivtives of functions on smooth mnifolds. In mny pplictions, c = c 0 is geodesic segment. Then c is smooth nd hence the term involving the (possible) breks c (t i ) vnishes. Furthermore, c vnishes by definition, nd therefore the integrl on the right hnd side vnishes s well. Corollry 1.4. Let c : [, b] M be geodesic segment nd H : U [, b] M piecewise smooth 1-prmeter vrition of c. Let V be the vrition field of H nd set c s := H(s, ) nd L(s) := L(c s ). Then s L(0) = cos α V () cos β V (b), where α = (c (), V ()) nd β = ( c (b), V (b)).

5 Vritionl Theory of Geodesics Geodesic Segments s Criticl Points. In the First Vrition Formul 1.3 it is not required tht the vrition be proper. If we restrict to proper vritions, then it turns out tht the curve c is criticl point of the length functionl if nd only if c is geodesic segment. Theorem 1.5. A regulr piecewise smooth curve c : [, b] M is geodesic if nd only if c hs constnt (nonzero) speed nd the first vrition of rc length vnishes for every proper vrition of c. Proof. If c is geodesic, then c is smooth nd of constnt speed v. ow v 0 since c is regulr. By definition c = 0, nd hence the first vrition of rc length vnishes for every proper vrition of c. We ssume now tht c hs constnt speed v 0 nd tht the first vrition of rc length vnishes for every proper vrition of c. Let = t 0 < t 1 <... < t k = b be subdivision of [, b] such tht c is smooth on [t i 1, t i ], 1 i k. Fix i {1,..., k} nd let ϕ : R R be smooth function which is positive on (t i 1, t i ) nd 0 otherwise. Let V = ϕ c. Then V is piecewise smooth vector field long c with V (t) = 0 for t [, t i 1 [t i, b]. In prticulr, there is proper vrition (c s ) of c with vrition field V. By the first vrition formul nd our ssumption on c we hve 0 = s L(0) = ti t i 1 ϕ D t c 2 dt. Hence c (t) = 0 for ll t [t i 1, t i ]. Since i ws rbitrry, we conclude tht c [t i 1, t i ] is geodesic, 1 i k. It remins to show c (t i 0) = c (t i +0) for ech i {1,..., k 1}. To tht end, fix such n i nd let E be the prllel field long c with E(t i ) = c (t i 0) c (t i +0). Let ϕ : R R be smooth function which is positive on (t i 1, t i+1 ) nd 0 otherwise nd set V = ϕ E. Let (c s ) be proper vrition of c with vrition field V. Since the first vrition of rc length of ny proper vrition of c vnishes nd c is geodesic on [t i 1, t i ] nd [t i, t i+1 ], we get 0 = s L(0) = V (t i ), c (t i 0) c (t i + 0) = ϕ(t i ) c (t i 0) c (t i + 0) 2. Hence c (t i 0) = c (t i + 0), 1 i k 1, nd hence c is geodesic. A geodesic segment c : [, b] M with c = 0 is not criticl point of the length functionl on the spce Ω ps (M) of ll piecewise smooth curves on M. For exmple, for the smooth vrition H(s, t) = c( + (1 s) (t )) we hve L(c s ) = (1 s) L(c) nd hence s L(0) = L(c) 0. The point is tht boundry conditions hve to be imposed, in the cse of Theorem 1.5 the boundry condition is tht the end points c() = p nd c(b) = q be fixed by the vrition. Other boundry conditions re discussed below.

6 6 Bsic Differentil Geometry 1.3. Second Vrition of Arc Length. From now on we ssume tht c : [, b] M is geodesic with speed v 0. We hve shown tht c is criticl point of the length functionl: The first vrition of rc length is zero for ny proper vrition of c. We now discuss the second derivtive of L. Second Vrition Formul 1.6. Let H : U [, b] M be piecewise smooth 2-prmeter vrition of c. Set c r,s := H(r, s, ), V = r H(0, 0, ), W = s H(0, 0, ), nd let ˆV nd Ŵ be the prt of V nd W perpendiculr to c. Then L(r, s) = L(c r,s ) is smooth in (r, s) nd r,sl(0, 2 0) = 1 { v D r s H, c b b + ( ˆV, Ŵ R( ˆV ) }, c )c, Ŵ dt = 1 { v D r s H, c + ˆV, Ŵ b b k 1 + ˆV (t i ), Ŵ (t i) ˆV + R( ˆV, c )c, Ŵ dt }, where the subdivision = t 0 <... < t k = b of [, b] is chosen such tht H is smooth on the sets [t i 1, t i ] nd ˆV (t i ) := ˆV (t i 0) ˆV (t i + 0). Proof. By diminishing the size of U if necessry, we cn ssume c r,s(t) 0 for ll (r, s, t) in U [t i 1, t i ], 1 i k. Then c r,s(t) is smooth in (r, s, t) nd we get D t s H, t H t H { D r D t s H, t H = + t H 2 r,sl(0, 0) = r { } dt r=s=0 D t s H, D t r H t H D t s H, t H D t r H, t H t H 3 dt} r=s=0. ow for the second term II nd third term III on the right hnd side we hve II III = 1 v ˆV, Ŵ dt. In the first term I, the denomintor is v nd hence I = 1 { v D t D r s H, t H dt + R( r H, t H) s H, t H dt For the second term I 2 on the right hnd side we hve I 2 = 1 v R(V, c )c, W dt = 1 v R( ˆV, c )c, Ŵ dt. } r=s=0.

7 Vritionl Theory of Geodesics 7 Since c is geodesic, D t t H(0, 0, ) = 0, hence the first term I 1 on the right hnd side bove is I 1 = 1 v t D r s H, t H dt = 1 k v D r s H, c t i = 1 t i 1 v D r s H, c b. This is the first of the sserted formuls. As for the second, we note tht t ˆV, Ŵ = ˆV, Ŵ + ˆV, Ŵ. ote tht the term D r s H, c b depends on the chosen vrition, not only on the tngent vectors V nd W to the vrition. This is due to the fct tht the geodesic segment c : [, b] M is not criticl point of the length functionl on the spce of ll piecewise smooth curves on M. The index form of c is defined s I(V, W ) : = 1 { ( v ˆV, Ŵ R( ˆV }, c )c, Ŵ ) dt = 1 v { k 1 ˆV (t i ), Ŵ (t i) ˆV + R( ˆV, c )c, Ŵ dt }, where the nottion is s in the Second Vrition Formul 1.6. The index form depends only on V nd W, not on the vrition defining V nd W nd is symmetric biliner form on the spce of piecewise smooth vector fields long c. Theorem 1.7. Suppose c : [, b] M is geodesic segment with speed c = v 0. Then for ny proper 2-prmeter vrition H of c 2 r,sl(0, 0) = I(V, W ). Tht is, the index form is the second derivtive of the length functionl L t c in the spce of piecewise smooth curves joining p = c() nd q = c(b). 2. Boundry Conditions So fr we discussed only one boundry condition, the fixed end point condition. There re other interesting boundry conditions. For exmple, for given submnifolds M nd M b of M, we my consider the condition c() M nd c(b) M b. The fixed end point cse is the specil cse where M nd M b re points. Another importnt exmple is the periodicity condition c(b) = c(). In generl, boundry condition is defined by submnifold of M M nd the boundry condition is (c(), c(b)). In the first exmple mentioned bove, = M M b ; in the second, is the digonl in M M. We fix boundry condition M M. We sk for criticl points of the length functionl L on the spce Ω ps of piecewise smooth curves c : [, b] M with (c(), c(b)). To tht end, we sy tht vrition (c s ) of such curve c is n -vrition if c s Ω ps for ll s. We sy tht c Ωps is criticl point of the length functionl L on Ω ps if the first vrition of rc length vnishes for ech

8 8 Bsic Differentil Geometry -vrition of c. ote tht proper vritions re -vritions for ny boundry condition. Hence by Theorem 1.5, criticl points of the length functionl L on Ω ps re geodesic segments. ow the following chrcteriztion is immedite from the First Vrition Formul. Theorem 2.1. Suppose c Ω ps is geodesic segment with c 0. Then c is criticl point of the length functionl L on Ω ps if nd only if (c (), c (b)) is perpendiculr to in (c(), c(b)). In the first exmple = M M b, this just mens tht c () is perpendiculr to M in c() nd tht c (b) is perpendiculr to M b in c(b). In the specil cse of fixed end points the ltter conditions re empty. In the second exmple, the periodic boundry condition, where is the digonl in M M, this mens tht c is closed geodesic; tht is c (b) = c (). ow we ssume tht c is geodesic segment with c = 0 nd (c (), c (b)) perpendiculr to in (c(), c(b)). By Theorem 2.1 this mens tht C is criticl point of the length functionl L on Ω ps. We observe tht for n -vrition of c, the first term in the Second Vrition Formul is precisely the second fundmentl form S of in the point (p, q) := (c(), c(b)) in the direction of the norml vector ( c (), c (b)). Using the nottion in the Second Vrition Formul 1.6, we get the following result. Theorem 2.2. Suppose c is geodesic segment with speed c = v 0 nd (c (), c (b)) perpendiculr to in (c(), c(b)). Then for ny 2-prmeter vrition H of c we hve 2 r,sl(0) = 1 v { S ( (V (), V (b)), ( c (), c (b)) ) + ( ˆV, Ŵ R( ˆV ) }, c )c, Ŵ dt = 1 { S v ( (V (), V (b)), ( c (), c (b)) ) + ˆV, Ŵ b k 1 + ˆV (t i ), Ŵ (t i) ˆV + R( ˆV, c )c, Ŵ dt }. In the first exmple = M M b, the second fundmentl form S of in (c(), c(b)) is the sum of the second fundmentl forms S of M in c() nd S b of M b in c(b). Hence S ( (V (), V (b)), ( c (), c (b)) ) = S b ( V (b), c (b) ) S ( V (), c () ). In the second exmple, the periodic boundry condition, we observe tht the digonl is totlly geodesic in M M. Hence in this cse, the first term in the Second Vrition Formul vnishes for ll -vritions.

9 Vritionl Theory of Geodesics 9 3. Energy Versus Arc Length The length of piecewise smooth curve c : [, b] M is invrint under reprmeteriztion. In considertions of more nlytic nture this is disdvntge nd it is better to use the energy of c, (3.1) E(c) = 1 2 c (t) 2 dt. One dvntge is cler right from the definition: Wheres the norm s function on the vector spce T p M, p M, is not smooth in the zero-vector 0 p T p M, the squre 2 of the norm is. A disdvntge of the energy functionl is its less geometric nture. The energy is not invrint under reprmeteriztion, this is the hert of the mtter here. In fct, by the Cuchy-Schwrz Inequlity we hve (3.2) L 2 (c) 2(b ) E(c) with equlity if nd only if c hs constnt speed, c (t) = v = const. As in the cse of rc length, there re formuls for the first nd second vrition of energy. These formuls re very similr to, but somewht simpler thn the ones for the first nd second vrition of rc length. First Vrition of Energy 3.1. Let c : [, b] M be piecewise smooth nd H : U [, b] M piecewise smooth 1-prmeter vrition of c. Let V be the vrition field of H nd set c s := H(s, ), E(s) := E(c s ). Then E : U R is smooth nd s E(0) = V, c b k 1 + V (t i ), c (t i ) V, D t c dt, where the subdivision = t 0 <... < t k = b of [, b] is such tht c is smooth on the sets [t i 1, t i ] nd c (t i ) := c (t i 0) c (t i + 0). We leve the proof of this formul nd of the next ssertions s n exercise. They consist of simplifictions of the proofs of the corresponding sttements for the length functionl. ote tht it is not necessry to ssume tht c hs constnt nd nonzero speed. As result, we hve the following more elegnt chrcteriztion of geodesics. Theorem 3.2. A piecewise smooth curve c : [, b] M is geodesic if nd only if the first vrition of energy vnishes for ny proper vrition of c.

10 10 Bsic Differentil Geometry Second Vrition of Energy 3.3. Let H : U [, b] M be piecewise smooth 2-prmeter vrition of c. Set c r,s = H(r, s, ), V = r H(0, 0, ), W = s H(0, 0, ). Then E(r, s) = E(c r,s ) is smooth in (r, s) nd r,se(0, 2 0) = D r s H, c b b ( ) + V, W R(V, c )c, W dt = D r s H, c b + V, W b k 1 + V (t i ), W (t i ) V + R(V, c )c, W dt, where the subdivision = t 0 <... < t k = b of [, b] is such tht H is smooth on the sets [t i 1, t i ] nd V (t i ) := V (t i 0) V (t i + 0) Semi-Riemnnin metrics. In the cse of Semi-Riemnnin mnifolds, the energy functionl cn be considered on the spce of ll piecewise smooth curves, the length functionl cn be considered on the spce of piecewise smooth curves with constnt positive speed, c, c = const > 0. The formuls for the first nd second vrition of energy nd length remin the sme. Acknowledgment I would like to thnk Alexnder Lytchk for proof-reding. Mthemtisches Institut, Universität Bonn, Beringstrsse 1, D Bonn,