Moving frames and compatibility conditions for three-dimensional director fields
aa r X i v : . [ c ond - m a t . s o f t ] F e b Noname manuscript No. (will be inserted by the editor)
Moving frames and compatibility conditions forthree-dimensional director fields
Luiz C. B. da Silva · Efi Efrati
Received: date / Accepted: date
Abstract
The geometry and topology of the region in which a director fieldis embedded impose limitations on the kind of supported orientational order.These limitations manifest as compatibility conditions that relate the quanti-ties describing the director field to the geometry of the embedding space. Forexample, in two dimensions (2D) the splay and bend fields suffice to determinea director uniquely and must comply with one relation linear in the Gaussiancurvature of the embedding manifold. In 3D there are additional local fields de-scribing the director, i.e. fields available to a local observer residing within thematerial, and a number of distinct ways to yield geometric frustration. So farit was unknown how many such local fields are required to uniquely describea 3D director field, nor what are the compatibility relations they must sat-isfy. In this work, we address these questions directly. We employ the methodof moving frames to show that a director field in 3D is fully determined byfive local fields. These fields are shown to be related to each other and to thecurvature of the embedding space through six differential relations. As an ap-plication of our method, we characterize all uniform distortion director fields,i.e., directors for which all the local characterizing fields are constant in space,in manifolds of constant curvature. The classification of such phases has beenrecently provided for directors in Euclidean space, where the textures corre-spond to foliations of space by parallel congruent helices. For non-vanishingcurvature, we show that the pure twist phase is the only solution in the spherewhile in the hyperbolic space uniform distortion fields correspond to foliationsof space by (non-necessarily parallel) congruent helices. Further analysis of
L. C. B. da SilvaE-mail: [email protected]. EfratiDepartment of Physics of Complex Systems,Weizmann Institute of Science, Rehovot 7610001, IsraelE-mail: efi[email protected] Luiz C. B. da Silva, Efi Efrati the obtained compatibility fields is expected to allow to also construct newnon-uniform director fields.
Keywords
Liquid crystal · Director field · Compatibility · Geometricfrustration · Moving frame
Liquid crystals are a state of matter characterized by the presence of an ori-entational order but no, or only partial, positional order. In many cases, theordering can be described in terms of a unit vector field n , called the director[9,27]. Liquid crystals pervade our daily lives, from computer and smart-phonedisplays to optical switches enabling fast and efficient communication. In re-cent years, liquid crystals also found applications as controllable and responsivematerials [1,11,30,24], and similar phases were identified outside soft mattersystems, for example in the nematic order observed in Iron based supercon-ductors [8,29].The liquid crystalline orientational “texture” often manifests the shape andinteractions between its constituents. Elongated and straight constituents withsteric interactions favor the nematic phase in which the director’s orientationis uniform in space. In contrast, chiral constituents may favor a twisted direc-tor field, while elongated and curved constituents may favor a bent director.However, not all such locally preferred tendencies can be globally realized by adirector field in a finite domain. For example, the 2D straight nematic texturewith vanishing splay and bend cannot be realized on any open region on thesphere [16]. Here, the splay and bend of a director field n are given by s = ∇· n and b = k ( n · ∇ ) n k , respectively, and constitute the basic distortion modes ofany 2D director field. Similarly, the phase of constant non-vanishing bend andvanishing splay cannot be realized in the plane [15]. It is thus natural to askwhat local tendencies could be realized by a director texture, and converselyhow many such local descriptors are required to uniquely determine a texture.Recently, it was shown that any 2D director field is fully described by itsbend and splay fields, and that the values these scalar fields obtain for anyrealizable texture satisfy K = − b − s − n · ∇ s + n ⊥ · ∇ b [16], where K is theGaussian curvature of the surface S in which the field is embedded and n ⊥ is the field in S normal to n . The identification of the class of all admissibletextures also allowed addressing the notion of optimal compromise for unre-alizable frustrated states. These results are, however, presently limited to 2Dsystems. For 3D liquid crystals there are additional distortion fields, such asthe twist and saddle-splay that do not have corresponding fields in 2D systems.Moreover, the 3D geometry is associated with additional compatibility con-ditions; while for 2D Riemannian geometry there is only one local geometriccharge, in 3D there are three independent scalar Riemannian charges. Thus,the 3D case is expected to lead to a larger set of relations involving a greaternumber of fields. Presently, it is unknown how many fields are required to oving frames and compatibility conditions for three-dimensional director fields 3 uniquely determine a director field in 3D, nor how many relations these fieldsmust satisfy to correspond to a realizable texture.Many frustrated assemblies, in which the constituents locally favor an ar-rangement that cannot be globally realized, exhibit a super-extensive groundstate energy for isotropic domains; i.e. the energetic cost of the optimal com-promise in these systems increases faster than linearly with their mass. Re-cently, in was shown that the exact order and structure of the compatibilityconditions completely determines this super-extensive behavior and can beused to predict the exponent related to the super-extensive growth of theground state energy [14]. The purpose of the work presented here is to fur-ther advance recent efforts aimed at understanding and quantifying frustrationin 3D liquid crystals. We provide a definitive answer to the above questionsby writing explicitly the six differential relations that form the compatibilityconditions relating the five fields that uniquely determine a director field in3D. These six equations relate the fields and their derivatives to each otherand to the curvature tensor of the 3D manifold where the director field lives.As an application of our results, we also characterize all uniform distortionfields in the sphere and hyperbolic space, showing in particular that in hy-perbolic space uniform distortion fields also correspond to a foliation of spaceby (non-necessarily parallel) helices . Thus, together with the results of Ref.[28], we complete the characterization of uniform distortion fields for all the3D homogeneous and isotropic geometries. The present work joins ongoing efforts to better understand the underlyinggeometry of 3D director fields. Recent insightful interpretations of the basicdistortion modes of unit director fields in 3D identified these distortion modeswith distinct components of the director gradient, J = ∇ n [13,21]. The splaycorresponds to the trace of J , while the bend is a vector in the space perpendic-ular to n and thus contributes two degrees of freedom. The remaining modescontribute to the components of J in the two dimensional space normal to n ,and are traceless. The twist t = n · ( ∇ × n ) corresponds to the anti-symmetriccomponent, while the biaxial splay is identified with the remaining tracelesssymmetric structure and thus contributes two degrees of freedom as well. Thisyields a total count of six independent contributions to J [13,21]. However,the freedom in assigning a base to the space perpendicular to n eliminates oneof these to yield five total intrinsic fields that describe a director. We iden-tify these as the splay, bend, twist, saddle-splay and the relative orientationbetween the direction of the bend vector and the principal direction of thebiaxial splay.These local descriptors of the liquid crystalline order may be associatedwith non trivial reference values induced by the structure and relative inter- By a helix we mean a curve of constant curvature and constant torsion. Luiz C. B. da Silva, Efi Efrati actions of their constituents. Considering phases composed of identical con-stituents, it is natural to assume that these reference values will be uniform inspace and manifest the underling symmetry of their constituent. However, aswas recently shown in [28], the space of phases associated with such constantdescriptors, termed “uniform distortions”, is very limited, necessitating morecomplex textures. For example, chiral constituents favoring the unrealizableuniform double twist produce the Blue phase in which defect lines, separatingbiaxially twisted columns, are periodically arranged [9]. Similarly, achiral bentcore liquid crystals form chiral meso-phases displaying giant optical activity[2,12] and heliconical ordering [6]. The constituents in this case locally favor aphase of vanishing twist, splay and saddle-splay and a constant non-vanishingbend. Such a phase cannot be realized in Euclidean space and instead the sys-tem incorporates a twist in order to accommodate the uniform bend resultingin the observed heliconical phase [15].Focusing on uniform distortions Virga showed that all such textures cor-respond to foliations of the three dimensional Euclidean space by parallel he-lices [28]. His results relied on vector calculus, where the motion of the frame { n , n , n } is described in terms of the so-called connectors vector fields andthe compatibility conditions for the deformation modes associated with a di-rector then follow from the symmetry of the tensors n i ·∇ n j [28]. In particular,it was shown that the pure bend phase favored by bent core liquid crystals isindeed frustrated, and predicted the heliconical phase with uniform twist as aplausible compromise. For small enough domains, however, one might expectother non-uniform distortions to yield the optimal compromise [14].Similar arguments show that the attempted pure double twist phase result-ing in the blue phase is also frustrated in Euclidean space [28]. This attemptedphase, however, can be accommodated in a 3D spherical geometry of an ap-propriate radius [22]. Other examples of uniform distortion fields have beenrecently provided for all the eight Thurston geometries [20], where it is shownthat each pure mode of director deformation can fill space without frustrationfor at least one type of geometry.In this work, we seek to obtain the full compatibility conditions for 3Ddirector fields. Naturally, one may seek to exploit the same reasoning thatwas exploited to yield the compatibility conditions in 2D [16]. However, themethod employed there relies heavily on the existence of a natural orthogonalframe of coordinates such that the parametric curves are tangent to n and tothe perpendicular unit vector n ⊥ . This, however, could not be generalized to3D. A general field of an orthonormal triad in 3D cannot be associated withthe tangents of parametric curves. Instead, one needs to study the propertiesof the orthonormal triad field without resorting to coordinates; the mathe-matical formalism which achieves this is called the method of moving frames [7], also known as vielbein formalism in the context of relativity [5]. Givena 3D director field n = n and its two normals n and n = n × n onecan build the corresponding dual frame of differential forms which togetherwith the so-called connection forms describe the geometry of 3D space usingthe differential form structure equations. This formalism also allows for an in- oving frames and compatibility conditions for three-dimensional director fields 5 variant formulation of vector calculus operators, which means that quantitiesand energy functionals used in the description of 3D liquid crystals can berewritten as exterior differential systems, i.e., differential equations in termsof differential forms and operations defined on them. Though more abstractthan the vector calculus method [28], the approach based on differential formsallows us to obtain manageable equations and to investigate director fields inboth Euclidean and curved Riemannian spaces in an equal foot. This helpsin better understanding how the Euclidean space frustrates the existence ofcertain phases.When concluding the writing of this manuscript a parallel effort to obtainthe compatibility conditions using moving frames by Pollard and Alexandercame to our attention [18]. We briefly relate to the similarities and differencesbetween these works in the discussion section. Given a coordinate system ( x , . . . , x m ) on an open and connected set U ⊆ R m ,the corresponding vector fields tangent to the coordinate curves are denotedby { ∂∂x i } , while their dual fields (or covectors ) are denoted by d x i , i.e., whenapplied to a vector v = v i ∂∂x i (sum on repeated indices), we have d x i ( v ) = v i .The differential of a scalar function f is defined as d f = ∂f∂x i d x i and,consequently, d x i can be alternatively seem as the differential of the i -th co-ordinate function. From now on, a field of covectors p ∈ U η p ∈ ( T p U ) ∗ is called a differential 1-form , while a function is a . Notice that wecan write any 1-form as η = a i d x i for some scalar fields a i and that thereis an isomorphism between vector fields and 1-forms: a i d x i ↔ a i ∂∂x i . Giventwo differential 1-forms η and ω , we define the exterior product η ∧ ω as theanti-symmetric bilinear map ( η ∧ ω )( u, v ) = η ( u ) ω ( v ) − η ( v ) ω ( u ). We shallrefer to η ∧ ω as a differential 2-form . We can define the exterior derivative of a 1-form η = a i d x i as the 2-form d η = d a i ∧ d x i = ∂a i ∂x j d x j ∧ d x i . Thevector space of 2-forms are generated by { d x i ∧ d x j } ≤ i 2. More generally, a differential k -form is an anti-symmetric k -linear map and the corresponding vector space is generated by thebasis { d x i ∧ · · · ∧ d x i k } ≤ i < ···
0) and n = ( − sin φ cos θ, − sin φ sin θ, cos φ ), where θ = θ ( x, y, z ) and φ = φ ( x, y, z )are smooth functions on U ⊆ R . Computing their differential givesd n = d φ ( − sin φ cos θ, − sin φ sin θ, cos φ ) + d θ ( − cos φ sin θ, cos φ cos θ, φ d θ n + d φ n , d n = d θ ( − cos θ, − sin θ, 0) = − cos φ d θ n + sin φ d θ n , d n = d φ ( − cos φ cos θ, − cos φ sin θ, − sin φ ) + d θ (sin φ sin θ, − sin φ cos θ, − d φ n − sin φ d θ n . Therefore, the 1-forms η ji associated with { n , n , n } are η = cos φ d θ , η =d φ , and η = sin φ d θ . We leave as an exercise checking the validity of thestructure equations d η ij = η kj ∧ η ik .The 1-forms η ki are also known as connection forms since they determinethe connection coefficients of the covariant derivative. Indeed, given two vectorfields u = u i n i and v = v i n i in U , the covariant derivative of u in the directionof v , ∇ v u , can be written using moving frames as ∇ v u = (d u )( v ) = d( u k n k )( v ) = (cid:2) d u k ( v ) + u j η kj ( v ) (cid:3) n k . (3) The use of the same symbol for both the differential of a map between manifolds and theexterior derivative of a differential form is justified by the possibility of seeing the differentialas a vector-valued 1-form, see, e.g. Subsect. 2.8 of Ref. [7].oving frames and compatibility conditions for three-dimensional director fields 7 Therefore, the connection forms can be alternatively computed from the Levi-Civita connection ∇ by using the relation η kj ( v ) = h∇ v n j , n k i . In addition,given two tangent vectors u, v ∈ T p U , the inner product between them is g ( u i n i , v j n j ) = u i v j δ ij = u i v i = η i ( u ) η i ( v ). The metric g in U is then writtenas g = ( η ) + · · · + ( η m ) . It follows that the geometry of U ⊆ R m is entirelycontained in the sets of 1-forms { η i } and { η ji } .To accomplish the goal of doing differential geometry using moving frames,we should be able to compute differential operators using differential forms. Todo that, we need the Hodge star operator ⋆ , which takes k -forms to ( m − k )-forms. Geometrically, we proceed as follows. Given a k -form ω = ω ∧ · · · ∧ ω k ,where { ω i } is linearly independent, consider the k -dimensional vector subspace V of R m generated by the vectors { v , . . . , v k } associated with { ω , . . . , ω k } .We then pick a basis { v k +1 , . . . , v m } of the vector space V ⊥ orthogonal to V and consider ω k +1 , . . . , ω m , the 1-forms associated with the vectors of thisbasis. Then, we define ⋆ ω = ± λ ω k +1 ∧ · · · ∧ ω m , where λ is the k -volumeof the solid generated by { v i } ki =1 and the sign corresponds to the orientationof B = { v , . . . , v k , v k +1 , . . . , v m } , i.e., plus if B has the same orientation asthe canonical basis of R m and minus if otherwise. Finally, we compute ⋆ for ageneric linear form by demanding linearity. As an example, in R the Hodgestar operator acting on 1-forms gives ⋆ d x = d x ∧ d x , ⋆ d x = − d x ∧ d x ,and ⋆ d x = d x ∧ d x . Finally, the curl and divergence of n are associatedwith differential forms according to ∇ × ˆ n ↔ ⋆ (d η ) and ∇ · ˆ n = ⋆ [d( ⋆ η )] , (4)where η the 1-form dual to ˆ n . Director fields n in 2D are fully described by their bend b = k n × ∇ × n k andsplay s = ∇ · n . However, the splay and bend are not independent functionsand they are related to the curvature of the ambient surface by [16] − K = s + b + n · ∇ s − n ⊥ · ∇ b. In this section we provide an alternative proof for the 2D compatibility equa-tion via moving frames. But, first, we shall illustrate how the moving framemethod can be used to describe the geometry of surfaces.Let r : U → S ⊂ R be a surface and N its unit normal. If { n , n } is a fieldof orthonormal bases for the tangent planes, we then define a moving framein 3D as { n , n , n := N } along with its dual frame { η , η , η } . Since weare interested on the surface geometry, we shall restrict our attention to η i , η ji when applied to tangent vectors. Then, in this restricted setting it follows that ∀ v = v n + v n ∈ T p S, η ( v ) = 0 . Therefore, seeing η as a 2D differential form on S implies η = 0. Thus, the1-forms η , η , η can be written as a linear combination of η and η only, Luiz C. B. da Silva, Efi Efrati i.e., they can also be seen as differential forms on the surface. This process ofseeing η i and η ij as 2D differential forms can be rigorously justified by using r to pullback the 1-forms η i and η ji to U : the pullback of a k -form η is the k -form ω = r ∗ η defined by ω p ( v , . . . , v k ) = η r ( p ) (d r ( v ) , . . . , d r ( v k )). Now,since the pullback operation ∗ commutes with d and ∧ [3], the 1-forms r ∗ η i and r ∗ η ij satisfy the same structure equations as η i and η ij . Thus, with someabuse of notation, we simply write η i = r ∗ η i and η ij = r ∗ η ij , which finallyjustifies seeing η i and η ji as 1-forms over S = r ( U ) ( ).From the fact that η = 0 on S , it follows that d η = 0 on S . Then, thestructure equations in (2) imply that η ∧ η + η ∧ η = 0 . An important resultfor differential forms is the Cartan lemma [4,7], which says that if ω , ... , ω k are linearly independent 1-forms and if there exist 1-forms θ , ... , θ k such that P ki =1 ω i ∧ θ i = 0, then θ i = a ij ω j with a ji = a ij . Therefore, since the set { η , η } is linearly independent, from the Cartan lemma we may write η = a η + a η and η = a η + a η , a ji = a ij . (5)From d n = η n + η n = − ( η n + η n ), it follows that the coefficients a ij precisely describe the shape operator of S . Then, the mean ( H ) and Gaussian( K ) curvatures can be written as H = 12 tr( a ) = a + a K = det( a ) = a a − ( a ) . (6)It remains to find the interpretation of η . From η ( v ) = h∇ v n , n i , we seethat we can write η = η ( n ) η + η ( n ) η = κ g η + κ ⊥ g η , where κ g and κ ⊥ g are the geodesic curvatures of the integral curves of n and n , respectively.In addition, taking the exterior derivative provides the important relationd η = η ∧ η = − Kη ∧ η . This relation will be the key to finding thecompatibility equation for director fields in 2D.We have just seen that for a surface in 3D the intrinsic geometry is encodedin η , η , and η , while the extrinsic geometry comes from η and η . (Thesecond fundamental form II can be written as II = η i η i .) The equation d η = − Kη ∧ η in 2D indicates that for moving frames in a Riemannian manifold thesecond set of structure equations, Eq. (2), must be modified to account for thecurvature of the ambient manifold: For a 2D manifold with Gaussian curvature K = R , the structure equations associated with the 1-forms { η , η } and η are d η = η ∧ η , d η = η ∧ η , and d η − η k ∧ η k = d η = − Kη ∧ η . As an alternative to using pullbacks, we could consider a foliation of space by surfacesparallel to S spanning a region parametrized as R ( x , x , x ) = r ( x , x ) + x N ( x , x ).Since we are only interested on tangent directions, any dependence of η ji on η does notcontribute to the final result. In addition, following this idea, η is nothing but the differentialof the x -coordinate, which implies that d η = 0. As shown in the main text, this is the keyproperty allowing us to use the moving frame method to study the differential geometry ofsurfaces in space.oving frames and compatibility conditions for three-dimensional director fields 9 In general, for a moving frame { n i } mi =1 in a Riemannian manifold M m withcurvature tensor R ijkℓ = R ijkℓ , the structure equations are [4,7]d η i = η k ∧ η ik and d η ij − η kj ∧ η ik = − R ijkℓ η k ∧ η ℓ = − X k<ℓ R ijkℓ η k ∧ η ℓ , (7)where we used that the operation of raising and lowering indices is trivial sincethe metric coefficients associated with the moving frame are δ ij = h n i , n j i .Now, let ˆ n be a director field on a 2D Riemannian manifold ( M , h· , ·i ).We may introduce a moving frame { n := ˆ n , n := ˆ n ⊥ } along with its coframe { η , η } . As we have seen, we can write η = κ g η + κ ⊥ g η .On the one hand, the splay s = ∇ · ˆ n is computed as s = ⋆ d ⋆ η = ⋆ d η = ⋆ ( η ∧ η ) = κ ⊥ g . (8)On the other hand, the bend b = k ˆ n × ∇ × ˆ n k is b = (cid:13)(cid:13) ⋆ ( η ∧ ⋆ d η ) (cid:13)(cid:13) = (cid:13)(cid:13) ⋆ [ η ∧ ⋆ ( η ∧ η )] (cid:13)(cid:13) = (cid:13)(cid:13) ⋆ [ η ∧ ⋆ ( κ g η ∧ η )] (cid:13)(cid:13) = (cid:13)(cid:13) κ g η (cid:13)(cid:13) = κ g . (9)This last equation also shows that, in 2D, we may write b = −∇ · n = ⋆ d ⋆ η .In short, we have the following relation η = b η + s η . (10)Now we shall apply the findings above in order to write the compatibilityequation for 2D director fields as found in [16], but using moving frames. Theorem 1 (Compatibility condition in 2D) Let ˆ n be a director fieldwith splay s and bend b on a 2D manifold M with Gaussian curvature K .Then, − K = s + b + (ˆ n · ∇ ) s − (ˆ n ⊥ · ∇ ) b, (11) where ( v · ∇ ) is the directional derivative in the direction of v and h ˆ n , ˆ n ⊥ i = 0 .Proof The exterior derivative of η isd η = d( b η + s η ) = d b ∧ η + b d η + d s ∧ η + s d η = [(ˆ n ⊥ · ∇ ) b ] η ∧ η + b η ∧ η + [(ˆ n · ∇ ) s ] η ∧ η + s η ∧ η = (cid:2) (ˆ n · ∇ ) s − (ˆ n ⊥ · ∇ ) b + b + s (cid:3) η ∧ η . Now, using that d η = − K η ∧ η we deduce the desired equality. ⊓⊔ Inspired by the study of 2D director fields, the strategy in 3D will consist ofwriting the 1-forms η ji in terms of the deformation modes of a director field n and then from the structure equations associated with d η ji we will obtain thecompatibility equations.In 2D, there are two deformation modes (bend and splay), while in 3Dthere are 6 modes, which can be further reduced to 5. Indeed, as discussed inSect. 2, taking into account rotations that preserve the director n , the gradient ∇ n decomposes as [13,21] ∇ α n β = − n α b β + s δ αβ − n α n β ) + t ǫ αβγ n γ + ∆ αβ , (12)where the Greek indices indicate Cartesian coordinates and b = − n · ∇ n isthe bend vector , whose norm b (the bend ) gives the curvature of the integrallines of n , s = ∇ · n is the splay , t = n · ∇ × n is the twist , and ∆ ij are thecoefficients of the biaxial-splay [21].In 2D, the coefficients of the 1-form η are related to the geometry of theintegral curves of the director and its orthogonal field. Given an integral curveof n i in 3D, we can consider { n i , n i +1 , n i +2 } as a positive orthonormal movingtrihedron along it, e.g., for n we have { n , n , n } . The equations of motionof such a moving trihedron along the n i -integral curves are ∇ n i n i n i +1 n i +2 = κ i κ i − κ i ω i − κ i − ω i n i n i +1 n i +2 , (13)where κ i and κ i relate to the (geodesic) curvature function κ i as κ i ( s ) = p [ κ i ( s )] + [ κ i ( s )] and ω i relates to the torsion τ i as ω i ( s ) = τ i ( s ) − θ ′ ( s ),where θ is the angle between the (Frenet) principal normal and n i +1 [25].Thus, using the property η ji ( v ) = h∇ v n i , n j i , the 1-forms η ji when written inthe basis { η , η , η } are η = κ η − κ η + ω η η = κ η − ω η − κ η η = ω η + κ η − κ η . (14)The 1-forms η and η provide information about the gradient of the di-rector n , d n = η n + η n . The components dual to the director ˆn thencontains information about the bend vector b = −∇ n n = b n × ∇ × n , b = p ( κ ) + ( κ ) . The remaining components of d n can be decomposedinto an antisymmetric and a symmetric part, where the symmetric part can befurther decomposed into a trace and traceless operator. This decompositionprovides the twist t , splay s , and biaxial splay coefficients ∆ ij , respectively.Thus, from (cid:18) − κ ω − ω − κ (cid:19) = t (cid:18) − 11 0 (cid:19) + s (cid:18) (cid:19) + (cid:18) ∆ ∆ ∆ − ∆ (cid:19) , (15) oving frames and compatibility conditions for three-dimensional director fields 11 we can write t = − ( ω + ω ) , s = − ( κ + κ ) , ∆ = κ − κ ∆ = ω − ω . (16)By inverting these relations, we can finally rewrite η and η in Eq. (14) as η = − b ⊥ η + (cid:16) s ∆ (cid:17) η + (cid:18) − t ∆ (cid:19) η (17)and η = − b × η + (cid:18) t ∆ (cid:19) η + (cid:16) s − ∆ (cid:17) η , (18)where we write the bend vector as b = b ⊥ n + b × n .The compatibility conditions then come from the structure equations Ω = d η − η ∧ η , Ω = d η − η ∧ η , and Ω = d η − η ∧ η , where Ω ji are the curvature forms whose coordinates in the basis { η i ∧ η j } provide the coefficients of the curvature tensor as defined in Eq. (7). Together,the three structure equations provide the 3 = 81 coefficients of the curvaturetensor R ijkℓ . From Ω ji = − Ω ij and the fact that Ω ji is a differential form, itfollows that R ijkℓ = − R jikℓ = − R ijℓk reducing these to only 9 independententries. However, the first Bianchi identity (which is required to further reducethese to only 6 independent components) cannot be proved directly from theabove definition. Proving this identity requires differentiating d η i = η k ∧ η ik toobtain η k ∧ Ω ik = 0, from which follows that R ijkℓ = R kℓij (see [4] or [7], p.376). While these relations hold for connection forms that are obtained froma moving frame, there could be 1-forms η ji that would fail to satisfy theserelations. Such forms could not be the connection forms of a moving framein any Riemannian geometry. Thus, while for any compatible set of movingframes the Riemann curvature tensor contains only six independent entries,requiring the satisfaction of the first Bianchi identity yields three additionalnon-trivial compatibility conditions resulting in the following nine equations: R = − ( s + ∆ ) , − b ⊥ , − b ⊥ − s + t − s∆ − ( ∆ ) + 2 ω ∆ + κ b × ,R = − ( − t + ∆ ) , − b ⊥ , − b ⊥ b × − s ( − t + ∆ ) − ω ∆ − κ b × ,R = − ( − t + ∆ ) , + ( s + ∆ ) , + tb ⊥ − κ ∆ + 2 κ ∆ ,R = − ( t + ∆ ) , − b × , − b ⊥ b × − s ( t + ∆ ) − ω ∆ − κ b ⊥ ,R = − ( s − ∆ ) , − b × , − b × − s + t + s∆ − ( ∆ ) − ω ∆ + κ b ⊥ ,R = − ( s − ∆ ) , + ( t + ∆ ) , + tb × − κ ∆ − κ ∆ ,R = − κ , + ω , − ( b × + κ )( s + ∆ ) + ( b ⊥ + κ )( t + ∆ )++ b ⊥ ω − κ ω ,R = κ , + ω , + ( b ⊥ + κ )( s + ∆ ) − ( b × + κ )( − t + ∆ )++ b × ω − κ ω ,R = κ , + κ , − ( κ ) − ( κ ) − t ω − s − t + ( ∆ ) , (19) where f ,i = n i · ∇ f denotes the derivative of f in the direction of n i and wedenote ∆ = p ( ∆ ) + ( ∆ ) .The gradient of the director field can be written in terms of the deforma-tions modes b ⊥ , b × , s, t, ∆ , and ∆ . However, notice that by choosing n to beeither the normalized bend vector or an eigenvector of the biaxial splay implieswe have a Gauge freedom allowing us to set either b × = 0 or ∆ = 0. There-fore, this reduces the number of degrees of freedom from 6 to 5. In addition,the equations for the curvature tensor were written in terms of 9 functions,six of which can be written in terms of the deformation modes. Thus, theremaining three, κ , κ , and ω must be superfluous. We will prove this lastassertion in the next two subsections, where we divide the study into directorfields with either ∆ = ( ∆ ) + ( ∆ ) > ∆ = ( ∆ ) + ( ∆ ) = 0 on allpoints. In short, we will have six 6 compatibility equations in 5 functions.5.1 Director fields with non-vanishing biaxial splayLet us assume non-vanishing biaxial splay ( ∆ ) = ( ∆ ) + ( ∆ ) > 0. Then,using the equations for R ijkℓ we can compute the sum ∆ R + ∆ R ,which allows us to write κ as κ = − ∆ R + ∆ R ∆ + ∆ t , − ∆ s , + ∆ s , + ∆ t , ∆ −− ∆ ∆ , − ∆ ∆ , ∆ + ( ∆ ) , ∆ + t b ⊥ ∆ + b × ∆ ∆ . (20)Using the equations for R ijkℓ we can find ∆ R − ∆ R , which allowsus to write κ as κ = − ∆ R − ∆ R ∆ − ∆ t , + ∆ s , − ∆ t , + ∆ s , ∆ ++ ∆ ∆ , − ∆ ∆ , ∆ + ( ∆ ) , ∆ − t b ⊥ ∆ − b × ∆ ∆ . (21)Analogously, computing − ∆ R + ∆ R + ∆ R + ∆ R allows usto write ω as ω = ∆ R − ∆ R − ∆ R − ∆ R ∆ + ∆ ∆ , − ∆ ∆ , ∆ −− ∆ b × , + ∆ b ⊥ , − ∆ b ⊥ , + ∆ b × , ∆ + ( b ⊥ − b × ) ∆ − b ⊥ b × ∆ ∆ −− κ b ⊥ ∆ + b × ∆ ∆ + κ b ⊥ ∆ − b × ∆ ∆ . (22) oving frames and compatibility conditions for three-dimensional director fields 13 Now, substituting the expressions for κ and κ in the equation above, wefinally have ω = ∆ R − ∆ R − ∆ R − ∆ R ∆ + b ⊥ R + b × R ∆ ++ ∆ ∆ , − ∆ ∆ , ∆ − ∆ b × , + ∆ b ⊥ , − ∆ b ⊥ , + ∆ b × , ∆ − tb ∆ −− b × ∆ , − b ⊥ ∆ , + b ⊥ ∆ , + b × ∆ , ∆ + ( b ⊥ − b × ) ∆ − b ⊥ b × ∆ ∆ . (23)There are three other linearly independent combinations we can constructwith the equations for R ij and R ij . Indeed, using the equations for R ijkℓ we can compute R + R , R − R = 0, and ∆ R + ∆ R + ∆ R − ∆ R , which give R + R = − s , − b ⊥ , − b × , − b − s t − ∆ + κ b × + κ b ⊥ , (24)0 = R − R = − t , − b × , + b ⊥ , − st − κ b ⊥ + κ b × , (25)and s = ∆ [ R − R ] − ∆ [ R + R ]2 ∆ − [( b ⊥ − b × ) ∆ + 2 b ⊥ b × ∆ ]2 ∆ −− ∆ [ b ⊥ , − b × , ] + ∆ [ b × , + b ⊥ , ]2 ∆ − κ b ⊥ ∆ − b × ∆ ∆ − κ b ⊥ ∆ + b × ∆ ∆ . (26)Note we can set 0 = R − R since this is required by the symmetries ofthe curvature tensor R ijkℓ .Now, substituting κ , κ , and ω from Eqs. (20), (21), and (23) in thethree equations we just obtained, we obtain three differential equations of firstorder involving the deformations modes. In addition, if we also substitute κ , κ , and ω in the equations for R ij , we will obtain another set of threedifferential equations involving the deformations modes and their first andsecond derivatives.5.2 Director fields with vanishing biaxial splayNow, let us assume a vanishing biaxial splay ( ∆ ) = ( ∆ ) + ( ∆ ) = 0. Ifwe assume that b = 0, then from the equations of R ijkℓ we can compute b × R − b ⊥ R , which allows us to write κ as κ = b × R − b ⊥ R b + b × s , − b ⊥ t , b + b ⊥ b × , − b × b ⊥ , b −− ( t − s ) b × + 2 stb ⊥ b . (27) In addition, from the equations of R ijkℓ we can compute b ⊥ R − b × R ,which allows us to write κ as κ = b ⊥ R − b × R b + b × t , + b ⊥ s , b + b ⊥ b × , − b × b ⊥ , b ++ ( s − t ) b ⊥ + 2 stb × b . (28)Note that when ∆ = 0, we can write ω as a function of the deformationmodes { s, t, b ⊥ , b × } by substituting for κ and κ in the equation for R .Alternatively, we can get rid of ω by choosing n and n such that ω = 0.On the other hand, if b = 0, then there are some restrictions on thegeometry of the ambient manifold. Indeed, we straightforwardly conclude that R = R and R = − R , which by using the symmetry R = R allows us to deduce that R = R = 0. In this section, we provide a characterization of uniform distortion fields, i.e.,director fields n for which the deformation modes { s, t, b, ∆ , ∆ } are all con-stant, in manifolds of constant sectional curvature. As a consequence, it willfollow that no combination of values other than the pure twist phase exist inpositive curvature. For negative curvature, the examples of uniform distortionfields with s + 4 b = 4 and t = 0 provided in [20] are in fact the most gen-eral case under the assumption that the biaxial splay vanishes. However, ourresults will also imply that it is possible to have uniform distortion fields innegative curvature with non-vanishing biaxial splay and, as in the Euclideanspace, these phases correspond to foliations of space by helices.From now on, let us assume that we have a director field in a curved space M of constant curvature R . This means that the curvature tensor is givenby R ijkℓ = R ( δ ik δ jℓ − δ iℓ δ jk ) [23]: R < M is locally isometric to ahyperbolic space, R = 0 if M is locally isometric to the Euclidean space,and R > M is locally isometric to a 3-sphere. Let us also introducethe shorthand notation h D b , b i = ( b ⊥ − b × ) ∆ + 2 b ⊥ b × ∆ = b ∆ cos(2 φ )and h JD b , b i = ( b × − b ⊥ ) ∆ + 2 b ⊥ b × ∆ = b ∆ sin(2 φ ), where φ is the angleformed by the bend vector and the principal direction of the biaxial splay.(In other words, D and J denote the biaxial splay and the counterclockwise π -rotation acting as linear operators on the plane normal to the director field,respectively.)The results of this section can be summarized as follows Theorem 2 Let M be a manifold of constant sectional curvature R and let n be a director field in it with constant deformation modes { s, t, b ⊥ , b × , ∆ , ∆ } .(a) If R > , then s = b = ∆ = 0 and t = ± √ R is the only solution. oving frames and compatibility conditions for three-dimensional director fields 15 (b) If R = 0 , then b = s = t = 0 when ∆ = 0 . On the other hand, when ∆ = 0 , then s = 0 , t = ± ∆ , and φ = (2 k +1) π , k ∈ { , , , } , i.e., thebend vector b bisects the principal directions of the biaxial splay, where k = 0 or k = 2 if t = 2 ∆ and k = 1 , if t = − ∆ .(c) If R < , then t = 0 and s + 4 b = − R when ∆ = 0 . On the otherhand, when ∆ = 0 , then t = ± ∆ , s = − ∆ h D b , b i , and the deformationmodes are subjected to the restriction b ∆ ≥ q − R ∆ .In (b) and (c), the bend and biaxial splay are the free parameters describingthe families of solutions. In the next subsections we are going to provide a proof for this theorem byanalyzing the restrictions imposed by the compatibility equations on the valuesof the deformation modes. But, before that, let us discuss the implications onthe geometry of the integral curves of the director field.For R > 0, it is known that the vector field tangent to the fibers of theHopf fibration provides an example of a uniform distortion field [20,22]. Itturns out that this is the only possibility. Indeed, given any uniform distortionfield n on a manifold of constant positive curvature, the integral curves of n are geodesics. In addition, from the fact that s, ∆ , and ∆ all vanish, wededuce that any two integral curves are parallel to each other, from whichfollows that the fibration provided by the integral curves of n is locally a Hopffibration [17].For R ≤ 0, the integral curves of a uniform distortion field do not have tobe geodesics. In general, they are helices, i.e., curves with constant curvatureand torsion. Indeed, the equations of motion of the { n , n , n } are ∇ n n n n = κ κ − κ ω − κ − ω n n n = − b ⊥ − b × b ⊥ ω b × − ω n n n , where ω is constant and given by Eq. (23). We can obtain the Frenet frame { T = n , N , B } from { n , n , n } by a rotation of an angle θ on the normalplane. Then, we can write κ = κ cos θ , κ = κ sin θ , and θ ′ = τ − ω [25].Since κ i and ω are all constant, we deduce that κ and τ are also constant.As a consequence, the integral curves of n make a constant angle with theDarboux vector field w = ω n − κ n + κ n or w = τ T + κ B if we usethe Frenet frame: if s denotes the arc-length of the integral curves of n , then dd s h w , n i = h∇ n w , n i + h w , ∇ n n i = h τ ′ n + b ′ B + τ b N − τ b N , n i + h w , b N i = 0.Virga’s strategy to characterize the helicoidal phases in Euclidean space[28] consisted in investigating the behavior of the frame along a generic curvein space (not necessarily an integral curve). This gives rise to an operatorwhose eigenvector can be shown to be constant and, in addition, the integrallines of the director field precess around this fixed direction. We can providean alternative proof by showing that all integral curves of a uniform distortionfield n in a flat manifold have the same axis, i.e., we may show that v · ∇ w = 0for every direction v . Using Eqs. (13) and (16) to write some of the κ ji ’s and ω i ’s as functions ofthe deformation modes, we conclude that w , = ( b × b ⊥ − b ⊥ b × ) n + ( b ⊥ ω − ω b ⊥ ) n + ( b × ω − ω b × ) n = 0 , w , = [ b ⊥ ( t ∆ ) − b × ∆ ] n + ( ω ∆ + b ⊥ κ ) n + [ ω ( t ∆ ) + b × κ ] n , and w , = [ b × ( t − ∆ )+ b ⊥ ( s − ∆ )] n − [ ω ( t − ∆ )+ b ⊥ κ ] n +[ ω ( s − ∆ ) − b × κ ] n . On the one hand, for R = 0, substitution of the values of the deformationmodes of a uniform distortion field allows us to deduce that n i · ∇ w = 0, i = 1 , , 3. Therefore, n provides a foliation of space by parallel helices. On theother hand, for R < 0, we still have that w , = 0 and, therefore, w is paralleltransported along the integral curves of n . On the other hand, in general w , and w , do not vanish, implying that n provides a foliation of hyperbolic spaceby helices which are not necessarily parallel. In a hyperbolic space, we need todistinguish between three types of helices. First notice that a curve with zerotorsion is necessarily contained in a totally geodesic surface, i.e., locally thesurface is a copy of a hyperbolic plane of curvature R . There are three typesof planes curves with constant curvature b > 0: circles if b ∈ ( √− R , ∞ ),horocycles if b = √− R , and hypercycles if b ∈ (0 , √− R ) [19]. Therefore,depending on the values of the bend b , we have three families of helices inhyperbolic geometry.6.1 Uniform distortion fields with vanishing biaxial splay( ∆ ) = ( ∆ ) + ( ∆ ) = 0First, assume we have b ⊥ = b × = 0. Then, from the equations for R and R , it follows that R = ( t − s ) and st = 0. Consequently, s = 0 or t = 0and we finally conclude ∆ = 0 , b = 0 ⇒ (cid:26) s = 0 and t = ± √ R if R ≥ t = 0 and s = ± √− R if R ≤ . (29)In particular, in Euclidean space, ∆ = 0 and b = 0 imply that the directorfield is constant: d n ≡ ∆ = 0 but b = 0. From R = 0 and R = 0, wenecessarily have t = 0. From Eqs. (27) and (28) it follows κ = b × b (cid:18) R + s (cid:19) and κ = b ⊥ b (cid:18) R + s (cid:19) . (30)Substituting the expressions for κ , κ in the equation for R gives R = − s − ( R + s b ⊥ + b × b ⇒ b (cid:18) R + s (cid:19) (cid:18) b + R + s (cid:19) = 0 . (31) oving frames and compatibility conditions for three-dimensional director fields 17 Therefore, s = − R or s = − b − R .On the one hand, we see that if R ≥ 0, then there exists no solution with b = 0. On the other hand, if R < 0, we could equally have either s = − R or s = − b − R (there is is no sign obstruction for R < s = − b − R is allowed. Indeed, if it were s = − R , then substituting theexpressions for κ and κ above in the equations for R , R and summingthem would give2 R = − b − s R + s ⇒ R = − b − s − b − ( − R )4 ⇒ b = 0 . (32)This contradicts the assumption that b = 0. Finally, we conclude that ∆ = 0 , b = 0 ⇒ (cid:26) ∄ solution if R > t = 0 and s + 4 b = − R if R ≤ . (33)Notice that in the case R < 0, such as in hyperbolic space, the configurationwith ∆ = 0 and t = 0 becomes the trivial director field in Euclidean space inthe limit R → − .6.2 Uniform distortion fields with non-vanishing biaxial splay( ∆ ) = ( ∆ ) + ( ∆ ) > κ , κ , and ω are also constant and equal to κ = t b ⊥ ∆ + b × ∆ ∆ , κ = − t b ⊥ ∆ − b × ∆ ∆ , and ω = − tb ∆ − h JD b , b i ∆ . (34)Now, substituting κ and κ in Eq. (26), implies that the splay is given by s = − h D b , b i ∆ . (35)In addition, substituting κ , κ , and ω in Eqs. (24), (25), and R from Eq.(19), gives2 R = − b − s t − ∆ + t ( b × − b ⊥ ) ∆ + 2 b ⊥ b × ∆ ∆ , (36)0 = − st − t ( b ⊥ − b × ) ∆ + 2 b ⊥ b × ∆ ∆ , (37)and R = − s − t ∆ − t b ∆ + t ( b × − b ⊥ ) ∆ + 2 b ⊥ b × ∆ ∆ . (38)Notice that from the expression we got for the splay in Eq. (35), it followsthat Eq. (37) is redundant. Subtracting Eq. (36) from twice the last equationabove allows us to conclude that0 = (cid:18) b ∆ (cid:19) ( t − ∆ ) ⇒ t = ± ∆. (39) Remark 1 The equations for R and R in (19) provide no further con-straints. In fact, substituting κ , κ , and ω in R and in R respectivelygives[1+ b ∆ ][1 − t ∆ ][ b × ∆ − b ⊥ ∆ ] = 0 and [1+ b ∆ ][1 − t ∆ ][ b ⊥ ∆ + b × ∆ ] = 0 . (40)Now, taking into account that t = ± ∆ , which is obtained from R , thetwo equations above vanish identically.Let us write t = 2 δ∆ , δ = ± 1, and substitute for t and s in Eq. (36). Thus,2 R = − b − h D b , b i ∆ + δ h JD b , b i ∆ , (41)from which we find that2 R ∆ + h D b , b i ∆ + b ∆ + δ h JD b , b i = 0 . (42)Now, from the Cauchy-Schwarz inequality, it follows that |h JD b , b i| ≤ k JD b kk b k ≤ b ∆ ⇒ ≤ b ∆ + δ h JD b , b i . We immediately have the following conclusions:(a) If R > 0, then Eq. (42) is a sum of non-negative numbers. However, R ∆ > ∆ > R = 0, then we must have h D b , b i = 0 and also b ∆ + δ h JD b , b i = 0.It follows that in a space of vanishing curvature, such as the Euclideanspace, the splay s must vanish and the bend vector b bisects the principaldirections of the biaxial splay, i.e., φ = (2 k +1) π , k ∈ { , , , } , where k = 0 or k = 2 if t = 2 ∆ and k = 1 , t = − ∆ .It remains to further analyze uniform distortion director fields in hyperbolicgeometry, i.e., R < 0. Seeing Eq. (36) as a quadratic polynomial in t , itsdiscriminant is disc. = 4 ∆ (cid:18) b ∆ (cid:19) + 4 R . (43)Thus, the requirement that t ∈ R demands disc. ≥ 0, which implies1 + b ∆ ≥ r − R ∆ . (44)Note that if we choose ∆ ≥ √− R , then the above inequality imposes norestriction on the values of the bend b . oving frames and compatibility conditions for three-dimensional director fields 19 In the intrinsic approach materials are described only through quantities avail-able to an observer residing within the material [14]. These quantities may beassociated with some non-trivial locally preferred reference values that mani-fest the constituents’ shape and mutual interactions. We show that a collectionof five such scalar (and pseudoscalar) fields suffice to characterize the directortexture. These fields can be chosen to be the bend, splay, twist, saddle-splayand the relative orientation between the principal biaxial splay direction andthe bend direction.In two dimensions only two such fields suffice to uniquely prescribe a direc-tor field. The compatibility conditions in the two dimensional case amount toa single first order differential relation [16]. In three dimensions we obtainedsix differential relations. Three of first order, and three of second order. Thusthe system is of at most second order; it is presently unknown if the system canbe further reduced to yield a purely first order system or not. Understandingthe degree and structure of the compatibility conditions is important not fortaxonomical reasons, but rather as these determine the super-extensive rateat which energy accumulates when a frustrated phase grows in size [14].Though more abstract than an approach uniquely based on vector calculus,the method of moving frames allows us to obtain manageable equations andto investigate director fields in both Euclidean and curved Riemannian spacesin an equal foot. In particular it allows us to find all uniform distortion fieldsfor all isotropic homogeneous Riemannian manifolds.Considering the five characterizing fields as given quantities and solvingfor the corresponding director field may also be carried out as long as thecompatibility conditions are satisfied. The compatibility conditions, in turn,can be interpreted both in terms of a Lagrangian frame, where the fields aregiven in terms of the material coordinates, and an Eulerian frame in whichthe field are given explicitly in terms of the embedding space coordinates. It isimportant to note that the information contained in the two viewpoints is notequivalent. The Lagrangian description, which is more natural from a materialperspective, may be used to integrate the director field from knowledge aboutits local behavior. Such an approach is particularly useful for solving inversedesign problems [11], and for constructing new optimal textures. The Eulerianapproach is somewhat less natural as it assumes that the fields are given interms of the stationary embedding space coordinates, yet the director is un-known. For the two dimensional case the Eulerian approach allowed obtainingthe director field directly from the gradients of the bend and splay functions,provided that the magnitude of these gradients was large enough [16]. The fullanalysis of the Eulerian three dimensional case remains outside the scope ofthe present work.When finishing this manuscript, it came to our attention that a similar ap-proach to the one presented here was recently pursued by Pollard and Alexan-der [18]. There, the authors also develop the idea of using the moving framemethod to obtain the compatibility conditions for the deformation modes of a director field expressed in terms of the curvature tensor of the ambient man-ifold. In addition, they consider methods to reconstruct a 3D director fieldaccording to the data which is prescribed. Finally, they discuss the examplesconstructed by Sadoc et al [20] in the language of moving frames by exploitingthe fact that there exists an underlying Lie algebra structure associated withuniform distortion fields.While the choice of applications differs between our work and that of Pol-lard and Alexander, the main guiding principles and calculations of the compo-nents of the Riemann curvature tensor are similar, save two differences. First,Pollard and Alexander retain nine variables in their equations, whereas wechoose to reduced these to only five. Second, they assume the satisfaction ofthe first Binachi identity and thus obtain only six equations from the structureequations, instead of the system of nine equations obtained here in Eq. (19).Thus, while in both cases the compatibility conditions consist of six differentialrelations, the obtained equations are somewhat different. Acknowledgements This work was funded by the Israel Science Foundation grant no.1479 / 16. 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