3-Dimensional Schlaefli Formula and Its Generalization

3-Dimensional Sc hlaefli F orm ula and Its Generalization F eng L uo ∗ Dep artment of Mathematics Rut gers University Pisc ataway, NJ 08854, USA email: fluomath.r utgers.edu De dic ate d to the memory of Xiao-Song Lin Abstract. Several identities similar to the Schlaefli form ula are established for tetrahedra in a space of constant curv ature. Keywo rds: tetrahedron, dihedral angles, volume, lengths, and the cosine la w. Con ten ts 1 Int ro duction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 The main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3 A pro of of theore m 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . 4 3.1 The partial deriv atives ∂ a ij ∂ x ik and ∂ a ij ∂ x kl . . . . . . . . . . . . . . . 5 3.2 The partial deriv ative ∂ a ij ∂ x ij . . . . . . . . . . . . . . . . . . . . . 7 1 In tro duction One of the most impor tan t iden tities in low-dimensional geometry is the Sc hlae- fli formula. It states that for a tetrahedron in a constan t curv ature λ = ± 1 space, the volume V , the length x ij , and the dihedral angle a ij at the ij-th edge are r elated by ∗ W ork partially supported by the NSF. 2 F eng Luo ∂ V ∂ a ij = λ 2 x ij (1.1) where V = V ( a 12 , a 13 , a 14 , a 23 , a 24 , a 34 ) is a function of the angles. See for instance [5] or [1] for a pro of. V i V j V k V l a ij X ij b ij k Figure 1.1 In 3-dimensio n al p olyhedral geometry , a space is o b tained b y iso m etrica lly gluing tetrahedra along their codimension- 1 faces. The metric is determined by the edge lengths and the curv ature at an edge is 2 π less the sum o f dihedral angles at the edge. F rom this po int of view, t he Sc hlaefli formula relates the most imp orta nt geometric q uantities: the volume, the metric (=edge lengths) and the curv a ture (= dihedral angles) in a simple elegant iden tity . The Sc hlaefli formula pla ys a vital role in a v a riational principle for tria ngulated 3-ma nifolds. See fo r instance Regge’s work [6] on discr e te g eneral relativity . One consequence of (1.1) is that differential 1-fo rms X x ij da ij and X a ij dx ij (1.2) are c lo sed. W e may recov er the v olume fu nction V in (1.1) by integrating the 1-form P x ij da ij . Thus the closeness of the 1-forms in (1.2) essentially captures the Schlaefli formula. The basic problem in p olyhedra l geometry is to understand the relation- ship betw een the metric a nd its cur v ature . In the case o f tetra hedra, this prompts us to study the curv ature map K ( x ) = a sending the edg e length x = ( x 12 , x 13 , x 14 , x 23 , x 24 , x 34 ) to the dihedral angle a = ( a 12 , a 13 , a 14 , a 23 , a 24 , a 34 ). The Ja cobian matrix D ( K ) of the curv ature map is the 6 × 6 matrix [ ∂ a ij ∂ x rs ] 6 × 6 . The closeness of t he 1-forms in (1.2) is equiv alent to say that the Jacobian matrix D ( K ) is sy mmetr ic. It tur ns out the J a cobian matrix [ ∂ a ij ∂ x rs ] 6 × 6 enjoys many mo r e symmetries. O ne of the symmetry was discov ered b y E. Wigner [9] and T a ylor -W oo dward [8]. The purpose o f this pap er is to find the co m- 3-Dimensional Schlaefli F orm ula and Its Generalizatio n 3 plete set of symmetries of the Jacobian matrix D ( K ) of the curv ature map. These symmetries should hav e applications in 3-dimensional top olo gy and ge- ometry . In particular, the relationships b etw een the Jacobia n matrix D ( K ), the 6j sym b ols, the quantum 6j s ymbols, and the v olume conjecture are very attractive problems. See fo r insta nce the w or k of [8] and [7 ]. The c o mplete set of s ymmetries was discovered by us a few years ago. W e thank W a lter Neumann for suggesting us to wr ite it up fo r publication. This pap er is dedicated to the memory of Xiao-Song Lin who made impor- tant contributions to low-dimensional top ology . He was a gr e at collea gue a nd friend. The pap er is org a nized as follows. In § 2, w e state the ma in theorem. These theorems are prov ed in § 3. A more general v ersion of it inv olving complex v alued lengths a nd a ngles can be found in [4]. 2 The main theorem Let a tetrahedr on in S 3 or H 3 or E 3 hav e v ertices v 1 , v 2 , v 3 , v 4 . Let a ij and x ij be th e dihedral angle and the edg e length at the ij-th edge v i v j . W e consider the angle a ij as a function of the lengths x 12 , x 13 , x 14 , x 23 , x 24 , x 34 . Here is o ur main result. Theorem 2. 1. Define P ij r s = 1 sin( a ij ) s in( a rs ) ∂ a ij ∂ x rs . Then these quantities sat- isfy the fol lowing identities for any tetr ahe dr on in any of the thr e e ge ometries S 3 , H 3 , E 3 . The ind ic es i, j, k, l ar e assu me d to b e p airwise di stinct. (i) (Sch laefli) P ij r s = P r s ij . (ii) ([9], [8]) P ij kl = P ik j l = P il j k . (iii) P ij ik = − P ij kl cos a j k . (iv) P ij ij = P ij kl w ij wher e w ij = ( c ij c j k c ki + c ij c j l c li + c ik c j l + c il c j k ) / sin 2 ( a ij ) and c r s = co s( a r s ) . (v) P ij r s = P i ′ j ′ r ′ s ′ wher e { i, j } 6 = { r, s } and for a subset { a, b } ⊂ { i, j, k , l } , the set { a ′ , b ′ } is { i, j, k , l } − { a, b } . In the s paces S 3 and H 3 of constant curv a ture λ = ± 1 , a tetrahedro n is determined by its dihedra l ang les a ij . Th us the length x ij can b e co nsidered as a function of the angles. The s imila r theorem is, Theorem 2.2. Define R ij r s = 1 sin( √ λx ij ) sin( √ λx rs ) ∂ x ij ∂ a rs . Then these quantities satisfy the fol lowing identities for any tetr ahe dr on in spheric al a nd hyp erb olic ge ometries. L et the indic es i, j, k , l b e distinct. (i) (Schlaefli) R ij r s = R r s ij . 4 F eng Luo (ii) ([9], [8]) R ij kl = R ik j l = R il j k . (iii) R ij ik = R ij kl cos( √ λx il ) . (iv) R ij ij = R ij kl w ij wher e w ij = − c ij c ik c il − c j i c j k c j l + c ik c j l + c il c j k sin 2 ( √ λx ij ) , and c r s = co s( √ λx r s ) , (v) R ij r s = R i ′ j ′ r ′ s ′ wher e { i, j } 6 = { r, s } and for a subset { a, b } ⊂ { i, j, k , l } , the set { a ′ , b ′ } is { i, j, k , l } − { a, b } . W e remark that the matrices [ ∂ a ij ∂ x rs ] and [ ∂ x ij ∂ a rs ] are inv erse of each other when λ 6 = 0. Theor em 2.2 follows from theorem 2.1 by taking the dual. Indeed, in the spherical tetrahedr al case, the dual tetrahedron has dihedra l a ngle π − x ij and edge length π − a ij at the kl-th edge of the dual simplex. Thus, theor em 2.2 follo ws. The hyper bo lic tetrahedra case in theorem 2.2 can b e deduced from spherical case b y a nalytical cont inuation. Theorem 2.1 sugg ests tha t the matrix M D ( K ) M = [ P ij r s ] 6 × 6 where M is the diagonal matrix w ho se diago nal entries are 1 sin( a ij ) exhibits mor e symmetries than the Jacobian matrix D ( K ). Both theorems ar e sp ecial cases o f a complex v alued edg e-length and dihe- dral angle relation. This will b e discussed in [4]. 3 A pro of of theorem 2.1 W e need to reca ll the cosine law and its deriv a tive form in order to co mpute the Jacobian matrix [ ∂ a ij ∂ x rs ] effectively . Let K 2 = S 2 , or H 2 or E 2 be the space of constan t curv ature λ = 1 , − 1, or 0. Define a function S λ ( t ) a s follows. S 0 ( t ) = t ; S 1 ( t ) = sin( t ) and S − 1 ( t ) = sinh( t ). The sine law for a tria ngle of lengths l 1 , l 2 , l 3 and opp osite a ngles a 1 , a 2 , a 3 in K 2 can be stated as S λ ( l i ) sin( a i ) = S λ ( l j ) sin( a j ) (3.1) A different wa y to sta te the s ine law is that the express ion A ij k = sin( a i ) S λ ( l j ) S λ ( l k ) is symmetric in indices i, j, k wher e { i , j, k } = { 1 , 2 , 3 } . F or this reas on, w e call A ij k = A 123 the A-invariant of the triang le . Prop ositi on 3.1. ([2], [3]) L et a triangle in K 2 have inner angles a 1 , a 2 , a 3 and e dge lengths l 1 , l 2 , l 3 so t hat l i -th e dge is opp osite to the angle a i . Then 3-Dimensional Schlaefli F orm ula and Its Generalizatio n 5 (i) ∂ a i ∂ l j = − ∂ a i ∂ l i cos( a k ) wher e { i, j, k } = { 1 , 2 , 3 } , (ii) ∂ a i ∂ l i = S λ ( l i ) A 123 See [2 ] or [3] for a pro of. Let us intro duce some notatio ns b efore beginning the pr o of. The indices i, j, k , l are pairwise distinct, i.e., { i, j, k , l } = { 1 , 2 , 3 , 4 } . The face triangle ∆ v i v j v k will b e denoted ∆ ij k . The inner angle a t the vertex v k of the triang le ∆ ij k is denoted b y b k ij . The link at the v ertex v k , denoted b y Lk ( v k ), is a spherical triangle with edge lengths b k j i , b k il , b k lj and inner a ngles a ki , a kj , a kl so that a ki is opp osite to b k j l . The A- inv ar iant of the triangle ∆ ij k is denoted b y A ij k . In the calcula tio n b e low, we c onsider b i j k as a function of x r s ’s using the cosine law for the triangle ∆ ij k . By the definition, we hav e, ∂ b i j k ∂ x r s = 0 (3.2) if { r , s } is no t a subset o f { i, j, k } . The function a ij is considered a s a function of b r st ’s by the co sine law applied to either the link Lk( v i ) or Lk( v j ). In this wa y the dihedral angle a ij , when co nsidered as a function of the lengths x r s ’s, is a co mp o sition function. T o prove theorem 2.1, note that identit y (i) in theor em 2 .1 is the Schlaefli formula (1.2). Identit y (v) follows from identit y (iii). By symmetry , we only need to consider three partia l der iv atives: ∂ a ij ∂ x kl , ∂ a ij ∂ x ik and ∂ a ij ∂ x ij . 3.1 The partial deriv ativ es ∂ a i j ∂ x i k and ∂ a i j ∂ x k l Consider the link Lk( v i ). Using pro p osition 3.1 (ii), the chain rule and (3.2), we hav e (see Fig . 2.1(a)), ∂ a ij ∂ x kl = ∂ a ij ∂ b i kl ∂ b i kl ∂ x kl = ∂ a ij ∂ b i kl S λ ( x kl ) A ikl . (3.3) Similarly , using Lk ( v j ), we hav e ∂ a ij ∂ x kl = ∂ a ij ∂ b j kl ∂ b j kl ∂ x kl = ∂ a ij ∂ b j kl S λ ( x kl ) A j kl . (3.4) 6 F eng Luo i j k l i j k l a ij b i kl b ik j x kl x ik b j kl b k jl b k ij (a) (b) Figure 2.1 Now we use the link Lk ( v j ) to find ∂ a ij ∂ x ik . By (3.2) and the chain r ule, we hav e ∂ a ij ∂ x ik = ∂ a ij ∂ b j ik ∂ b j ik ∂ x ik . (3.5) By prop osition 3.1 applied to Lk( v j ) a nd ∆ ij k , we see (3.5 ) is equa l to, − ∂ a ij ∂ b j kl cos( a j k ) S λ ( x ik ) A ij k . (3.6) Using (3.4), w e ca n wr ite (3 .6) as, − ∂ a ij ∂ x kl cos( a j k ) A j kl S λ ( x ik ) A ij k S λ ( x kl ) . (3.7) Now by the definition o f the A-inv ariant of triang les ∆ ij k and ∆ j k l (see Fig. 2.1(b)), we hav e, A j kl = S λ ( x j k ) S λ ( x kl ) sin( b k j l ) and A ij k = S λ ( x j k ) S λ ( x ik ) sin( b k ij ) . (3.8) Thu s (3.7) can b e simplified to − ∂ a ij ∂ x kl cos( a j k ) sin( b k j l ) sin( b k ij ) . (3 .9) By the s ine law a pplied to the spherica l tr iangle Lk( v k ), we see (3.9 ) is equal to ∂ a ij ∂ x ik = − ∂ a ij ∂ x kl cos( a j k ) sin( a ik ) sin( a kl ) . (3.10) This is equiv a le nt to identit y (iii), P ij ik = − P ij kl cos a j k . (3.11) Use the Schlaefli formula that P ij ik = P ik ij , we obtain from (3.11 ) 3-Dimensional Schlaefli F orm ula and Its Generalizatio n 7 − P ij kl cos( a j k ) = − P ik j l cos( a j k ) . This shows that P ij kl = P ik j l . (3.12) By symmetry , identit y (ii) holds for a ll indice s . 3.2 The partial deriv ativ e ∂ a i j ∂ x i j By (3.2), the c hain rule, we have, in the triangle Lk( v i ), ∂ a ij ∂ x ij = ∂ a ij ∂ b i j k ∂ b i j k ∂ x ij + ∂ a ij ∂ b i j l ∂ b i j l ∂ x ij . (3.13) Using prop osition 3.1, we see that (3.13 ) is equa l to ∂ a ij ∂ b i kl cos( a ik ) cos( b j ik ) S λ ( x j k ) A ij k + ∂ a ij ∂ b i kl cos( a il ) cos( b j il ) S λ ( x j l ) A ij l . (3.14) Using (3.3), we see (3.14) is equal to ∂ a ij ∂ x kl [cos( a ik ) cos( b j ik ) S λ ( x j k ) A ikl S λ ( x kl ) A ij k + c os( a il ) cos( b j il ) S λ ( x j l ) A ikl S λ ( x kl ) A ij l ] . (3.15) Using the sine law fo r triangle s ∆ ik l a nd ∆ ij l as in (3.8 ), we can rewrite (3.15) a s ∂ a ij ∂ x kl [cos( a ik ) cos( b j ik ) sin( b k il ) sin( b k ij ) + c os( a il ) cos( b j il ) sin( b l ik ) sin( b l ij ) ] . (3.16) Using the sine law in tria ngles L k( v k ) a nd Lk( v l ), we see tha t (3.16) is the same as ∂ a ij ∂ x kl [cos( a ik ) cos( b j ik ) sin( a kj ) sin( a kl ) + c os( a il ) cos( b j il ) sin( a lj ) sin( a lk ) ] . (3.17) = P ij kl [cos( a ik ) cos( b j ik ) sin( a kj ) sin( a ij ) + cos ( a il ) cos( b j il ) sin( a lj ) sin( a ij )] . (3.18) On the other ha nd, by the cosine law for the spherica l triangle Lk( v j ), we hav e cos( b j ik ) sin( a kj ) sin( a ij ) = cos a kj cos a ij + c os a lj . and 8 F eng Luo cos( b j il ) sin( a lj ) sin( a ij ) = cos a j l cos a ij + c os a j k . Substitute these into (3.18), we o btain ∂ a ij ∂ x ij = P ij kl [ c ij c j k c ki + c ij c j l c li + c ik c j l + c il c j k ] where c r s = cos( a r s ). This is the identit y (iv) since P ij ij = 1 sin 2 ( a ij ) ∂ a ij ∂ x ij . References [1] Alekseevskij, D. V.; Vin b erg, . B.; Solo dovnik ov, A . S. Geometry of spaces of constant curv ature. Geometry , I I, 1–138, Encyclopaedia Math. 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