1 Introduction

Multi linear formulation of differential geometry and matrix regularizations

Abstract.

We prove that many aspects of the differential geometry of embedded Riemannian manifolds can be formulated in terms of multi linear algebraic structures on the space of smooth functions. In particular, we find algebraic expressions for Weingarten’s formula, the Ricci curvature and the Codazzi-Mainardi equations.

For matrix analogues of embedded surfaces we define discrete curvatures and Euler characteristics, and a non-commutative Gauss–Bonnet theorem is shown to follow. We derive simple expressions for the discrete Gauss curvature in terms of matrices representing the embedding coordinates, and a large class of explicit examples is provided. Furthermore, we illustrate the fact that techniques from differential geometry can carry over to matrix analogues by proving that a bound on the discrete Gauss curvature implies a bound on the eigenvalues of the discrete Laplace operator.

1. Introduction

It is generally interesting to study in what ways information about the geometry of a differentiable manifold can be extracted as algebraic properties of the algebra of smooth functions . In case is a Poisson manifold, this algebra has a second (apart from the commutative multiplication of functions) bilinear (non-associative) algebra structure, the Poisson bracket. The bracket is compatible with the commutative multiplication via Leibniz rule, thus carrying the basic properties of a derivation.

On a surface , with local coordinates and , one can define

 {f,h}=1√g(∂f∂u1∂h∂u2−∂h∂u1∂f∂u2),

where is the determinant of the induced metric tensor, and one readily checks that is a Poisson algebra. Having only this very particular combination of derivatives at hand, it seems at first unlikely that one can encode geometric information of in Poisson algebraic expressions. Surprisingly, it turns out that many differential geometric quantities can be computed in a completely algebraic way, cp. Theorem 3.7 and Theorem 3.17. For instance, the Gaussian curvature of a surface embedded in can be written as

 (1.1) K=m∑j,k,l=1(12{{xj,xk},xk}{{xj,xl},xl}−14{{xj,xk},xl}{{xj,xk},xl}),

where are the embedding coordinates of the surface.

For a general -dimensional manifold , we are led to consider Nambu brackets [Nam73], i.e. multi-linear alternating -ary maps from to , defined by

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In the case of surfaces, our initial motivation for studying the problem came from matrix regularizations of Membrane Theory. Classical solutions in Membrane Theory are 3-manifolds with vanishing mean curvature in . Considering one of the coordinates to be time, the problem can also be formulated in a dynamical way as surfaces sweeping out volumes of vanishing mean curvature. In this context, a regularization was introduced replacing the infinite dimensional function algebra on the surface by an algebra of matrices [GH82]. If we let be a linear map from smooth functions to hermitian matrices, the main properties of the regularization are

 limα→∞||Tα(f)Tα(g)−Tα(fg)||=0, limα→∞∣∣∣∣∣∣1iℏα[Tα(f),Tα(h)]−Tα({f,h})∣∣∣∣∣∣=0,

where is a real valued function tending to zero as (see Section 4 for details), and therefore it is natural to regularize the system by replacing (commutative) multiplication of functions by (non-commutative) multiplication of matrices and Poisson brackets of functions by commutators of matrices.

Although we may very well consider , its relation to is in general not simple. However, the particular combination of derivatives in is expressed in terms of a commutator of and . In the context of Membrane Theory, it is desirable to have geometrical quantities in a form that can easily be regularized, which is the case for any expression constructed out of multiplications and Poisson brackets. For instance, solving the equations of motion for the regularized membrane gives sequences of matrices that correspond to the embedding coordinates of the surface. Since the set of solutions contains regularizations of surfaces of arbitrary topology, one would like to be able to compute the genus corresponding to particular solutions. The regularized form of (1.1) provides a way of resolving this problem.

The paper is organized as follows: In Section 2 we introduce the relevant notation by recalling some basic facts about submanifolds. In Section 3 we formulate several basic differential geometric objects in terms of Nambu brackets, and in Section 3.1 we provide a construction of a set of orthonormal basis vectors of the normal space. Section 3.2 is devoted to the study of the Codazzi-Mainardi equations and how one can rewrite them in terms of Nambu brackets. In Section 3.4 we study the particular case of surfaces, for which many of the introduced formulas and concepts are particularly nice and in which case one can construct the complex structure in terms of Poisson brackets.

In the second part of the paper, starting with Section 4, we study the implications of our results for matrix regularizations of compact surfaces. In particular, a discrete version of the Gauss-Bonnet theorem is derived in Section 4.1 and a proof that the discrete Gauss curvature bounds the eigenvalues of the discrete Laplacian is found in Section 4.4.

2. Preliminaries

To introduce the relevant notations, we shall recall some basic facts about submanifolds, in particular Gauss’ and Weingarten’s equations (see e.g. [KN96a, KN96b] for details). For , let be a -dimensional manifold embedded in a Riemannian manifold with . Local coordinates on will be denoted by , local coordinates on by , and we regard as being functions of providing the embedding of in . The metric tensor on is denoted by and the induced metric on by ; indices run from to , indices run from to and indices run from to . Furthermore, the covariant derivative and the Christoffel symbols in will be denoted by and respectively.

The tangent space is regarded as a subspace of the tangent space and at each point of one can choose as basis vectors in , and in this basis we define . Moreover, we choose a set of normal vectors , for , such that and .

The formulas of Gauss and Weingarten split the covariant derivative in into tangential and normal components as

 (2.1) ¯∇XY=∇XY+α(X,Y) (2.2) ¯∇XNA=−WA(X)+DXNA

where and , and , . By expanding in the basis one can write (2.1) as

 (2.3) ¯∇XY=∇XY+p∑A=1hA(X,Y)NA,

and we set . From the above equations one derives the relation

 (2.4) hA,ab =−¯g(ea,¯∇bNA),

as well as Weingarten’s equation

 (2.5) hA(X,Y)=¯g(WA(X),Y),

which implies that , where denotes the inverse of .

From formulas (2.1) and (2.2) one obtains Gauss’ equation, i.e. an expression for the curvature of in terms of the curvature of , as

 (2.6) g(R(X,Y)Z,V)=¯g(¯R(X,Y)Z,V)−¯g(α(X,Z),α(Y,V))+¯g(α(Y,Z),α(X,V)),

where . As we shall later on consider the Ricci curvature, let us note that (2.6) implies

 (2.7) Rpb=gpdgac¯g(¯R(ec,ed)eb,ea)+p∑A=1[(WA)aa(WA)pb−(W2A)pb]

where is the Ricci curvature of considered as a map . We also recall the mean curvature vector, defined as

 (2.8) H=1np∑A=1(trWA)NA.

3. Nambu bracket formulation

In this section we will prove that one can express many aspects of the differential geometry of an embedded manifold in terms of a Nambu bracket introduced on . Let be an arbitrary non-vanishing density and define

 (3.1) {f1,…,fn}=1ρεa1⋯an(∂a1f1)⋯(∂anfn)

for all , where is the totally antisymmetric Levi-Civita symbol with . Together with this multi-linear map, is a Nambu-Poisson manifold.

The above Nambu bracket arises from the choice of a volume form on . Namely, let be a volume form and define via the formula

 (3.2) {f1,…,fn}ω=df1∧⋯∧dfn.

Writing in local coordinates, and evaluating both sides of (3.2) on the tangent vectors gives

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To define the objects which we will consider, it is convenient to introduce some notation. Let be the embedding coordinates of into , and let denote the components of the orthonormal vectors , normal to . Using multi-indices and we define

 {f,→xI}≡{f,xi1,xi2,…,xin−1} {f,→nIA}≡{f,ni1A,ni2A,…,nin−1A},

together with

 ∂→a→xI≡(∂a1xi1)(∂a2xi2)⋯(∂an−1xin−1) Missing or unrecognized delimiter for \big ¯gIJ≡¯gi1j1¯gi2j2⋯¯gin−1jn−1 g→a→c≡ga1c1ga2c2⋯gan−1cn−1.

We now introduce the main objects of our study

 (3.3) PiJ =1√(n−1)!{xi,→xJ}=1√(n−1)!εa→aρ(∂axi)(∂→a→xJ) (3.4) SiJA Missing or unrecognized delimiter for \big (3.5) TIjA Missing or unrecognized delimiter for \big

from which we construct

 (3.6) (P2)ik =PiIPkJ¯gIJ (3.7) (BA)ik =PiI(TA)Jk¯gIJ (3.8) (SATA)ik =(SA)iI(TA)Jk¯gIJ.

By lowering the second index with the metric , we will also consider , and as maps . Note that both and can be written in terms of Nambu brackets, e.g.

 TIjA=(−1)n√(n−1)![{→xI,njA}+{→xI,xk}¯ΓjklnlA].

Let us now investigate some properties of the maps defined above. As it will appear frequently, we define

 (3.9) γ=√gρ.

It is useful to note that (cp. Proposition 3.3)

 γ2=m∑i,j,I,J=11n!¯gij{xi,→xI}¯gIJ{xj,→xJ},

and to recall the cofactor expansion of the inverse of a matrix:

Lemma 3.1.

Let denote the inverse of and . Then

 (3.10) ggba=1(n−1)!εaa1⋯an−1εbb1⋯bn−1ga1b1ga2b2⋯gan−1bn−1.
Proposition 3.2.

For it holds that

 (3.11) P2(X)=γ2¯g(X,ea)gabeb (3.12) BA(X)=−γ2¯g(X,¯∇aNA)gabeb (3.13) SATA(X)=γ2(detWA)¯g(X,¯∇aNA)habAeb,

and for one obtains

 (3.14) P2(Y)=γ2Y (3.15) BA(Y)=γ2WA(Y) (3.16) SATA(Y)=−γ2(detWA)Y.
Proof.

Let us provide a proof for equations (3.11) and (3.14); the other formulas can be proved analogously.

 P2(X) =PiIPjJ¯gIJ¯gjkXk∂i=εa→aεc→cρ2(n−1)!(∂axi)(∂→axI)(∂cxj)(∂→cxJ)¯gIJ¯gjkXk∂i Missing or unrecognized delimiter for \big =γ2gac(∂axi)(∂cxj)¯gjkXk∂i=γ2¯g(X,ec)gcaea.

Choosing a tangent vector gives immediately that . ∎

For a map we denote the trace by and for a map we denote the trace by .

Proposition 3.3.

It holds that

 (3.17) 1nTrP2 =γ2 (3.18) TrBA =γ2trWA (3.19) 1nTrSATA =−γ2(detWA).
Remark 3.4.

For a hypersurface (with normal ) in ,

 (3.20) detW =(−1)n{xi1,…,xin}{ni1,…,nin}{xk1,…,xkn}{xk1,…,xkn} =1γn!εi1⋯ini{ni1,…,nin}ni,

the signed ratio of infinitesimal volumes swept out on (by ), resp (which can easily be obtained directly by simply writing out the determinant of the second fundamental form, ); in fact, all the symmetric functions of the principal curvatures are related to ratios of products of two Nambu brackets (cp. the paragraph after Proposition 3.11). Namely, the ’th symmetric curvature is given by

 (3.21) (−1)k{xi1,…,xin}{ni1,…,nik,xik+1,…,xin}{xk1,…,xkn}{xk1,…,xkn}.

A direct consequence of Propositions 3.2 and 3.3 is that one can write the projection onto , as well as the mean curvature vector, in terms of Nambu brackets.

Proposition 3.5.

The map

 (3.22) γ−2P2=nTrP2P2:TM→TΣ

is the orthogonal projection of onto . Furthermore, the mean curvature vector can be written as

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Proposition 3.2 tells us that equals the Weingarten map , when restricted to . What is the geometrical meaning of acting on a normal vector? It turns out that the maps also provide information about the covariant derivative in the normal space. If one defines through

 DXNA=p∑B=1(DX)ABNB

for , then one can prove the following relation to the maps .

Proposition 3.6.

For it holds that

 (3.23) Missing or unrecognized delimiter for \big
Proof.

For a vector , it follows from Weingarten’s formula (2.2) that

 (DX)AB=¯g(¯∇XNA,NB).

On the other hand, with the formula from Proposition 3.2, one computes

 ¯g(BB(NA),X) =−γ2¯g(NA,¯∇aNB)gabgbcXc=−γ2¯g(NA,¯∇XNB) =−γ2(DX)BA=γ2(DX)AB.

The last equality is due to the fact that is a covariant derivative, which implies that . ∎

Thus, one can write Weingarten’s formula as

 (3.24) γ2¯∇XNA=−BA(X)+p∑B=1¯g(BB(NA),X)NB,

and since Gauss’ formula becomes

 (3.25) ¯∇XY=∇XY+1γ2p∑A=1¯g(BA(X),Y)NA.

Let us now turn our attention to the curvature of . Since Nambu brackets involve sums over all vectors in the basis of , one can not expect to find expressions for quantities that involve a choice of tangent plane, e.g. the sectional curvature (unless is a surface). However, it turns out that one can write the Ricci curvature as an expression involving Nambu brackets.

Theorem 3.7.

Let be the Ricci curvature of , considered as a map , and let denote the scalar curvature. For any it holds that

 (3.26) R(X)=1γ4(P2)ik(P2)lm¯RijklXj∂m+1γ4p∑A=1[(TrBA)BA(X)−B2A(X)] (3.27) R=1γ4(P2)ik(P2)jl¯Rijkl+1γ4p∑A=1[(TrBA)2−TrB2A(X)],

where is the curvature tensor of .

Proof.

The Ricci curvature of is defined as

 Rpb=gacgpdg(R(ec,ed)eb,ea)

and from Gauss’ equation (2.6) it follows that

Since one obtains

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and as for any , and , one has

 R(X)=gacgpd¯g(¯R(ec,ed)eb,ea)Xbep+1γ4p∑A=1[(TrBA)BA(X)−B2A(X)].

By expanding the first term as

 gac gpdXb¯Rijkl(∂axi)(∂bxj)(∂cxk)(∂dxl)(∂pxm)∂m =1g2(n−1)!2εp→pεd→dg→p→dεa→aεc→cg→a→cXb¯Rijkl(∂axi)(∂bxj)(∂cxk)(∂dxl)(∂pxm)∂m =…=1γ4(P2)ik(P2)lm¯RijklXj∂m

one obtains the desired result. ∎

3.1. Construction of normal vectors

The results in Section 3 involve Nambu brackets of the embedding coordinates and the components of the normal vectors. In this section we will prove that one can replace sums over normal vectors by sums of Nambu brackets of the embedding coordinates, thus providing expressions that do not involve normal vectors.

It will be convenient to introduce yet another multi-index; namely, we let consist of indices all taking values between and .

Proposition 3.8.

For any value of the multi-index , the vector

 (3.28) Zα=1γ(n!√(p−1)!)¯gijεjk1⋯knα{xk1,…,xkn}∂i,

where is the Levi-Civita tensor of , is normal to , i.e. for . For hypersurfaces (), equation (3.28) defines a unique normal vector of unit length.

Proof.

To prove that are normal vectors, one simply notes that

 γ(n!√(p−1)!)¯g(Zα,ea) =1ρεa1⋯anεjk1⋯knα(∂axj)(∂a1xk1)⋯(∂anxkn)=0,

since the indices can only take on different values and since is contracted with which is completely antisymmetric in . Let us now calculate when . Using that1

 εik1⋯knεil1⋯ln=δ[l1[k1⋯δln]kn]

one obtains

 |Z|2 =1γ2n!2¯gl1l′1⋯¯glnl′nεik1⋯knεil1⋯ln{xk1,…,xkn}{xl′1,…,xl′n} =1γ2n!2¯gl1l′1⋯¯glnl′nδ[l1[k1⋯δln]kn]{xk1,…,xkn}{xl′1,…,xl′n} =1γ2n!{xl1,…,xln}¯gl1l′1⋯¯glnl′n{xl′1,…,xl′n} =1γ2n!(n−1)!TrP2=1γ2n!(n−1)!nγ2=1,

which proves that has unit length. ∎

If the codimension is greater than one, defines more than non-zero normal vectors that do not in general fulfill any orthonormality conditions. In principle, one can now apply the Gram-Schmidt orthonormalization procedure to obtain a set of orthonormal vectors. However, it turns out that one can use to construct another set of normal vectors, avoiding explicit use of the Gram-Schmidt procedure; namely, introduce

 Zβα=¯g(Zα,Zβ),

and consider it as a matrix over multi-indices and . As such, the matrix is symmetric (with respect to ) and we let denote orthonormal eigenvectors (i.e. ) and their corresponding eigenvalues. Using these eigenvectors to define

 ^Nα=EβαZβ

one finds that , i.e. the vectors are orthogonal.

Proposition 3.9.

For it holds that

 (3.29) ZδαZβδ=Zβα (3.30) Zαα=p.
Proof.

Both statements can be easily proved once one has the following result

 (3.31) ZiαZjα=¯gij−1γ2(P2)ij,

which is obtained by using that

 εkk1⋯knαεll1⋯lnα=(p−1)!(δ[l[kδl1k1⋯δln]kn]).

Formula (3.30) is now immediate, and to obtain (3.29) one notes that since it holds that , due to the fact that is proportional to the projection onto . ∎

From Proposition 3.9 it follows that an eigenvalue of is either 0 or 1, which implies that or , and that the number of non-zero vectors is . Hence, the non-zero vectors among constitute an orthonormal basis of , and it follows that one can replace any sum over normal vectors by a sum over the multi-index of . As an example, let us work out some explicit expressions in the case when .

Proposition 3.10.

Assume that and that all repeated indices are summed over. For any one has

 (3.32) p∑A=1(TrBA)BA(X)i=1(n−1)!2Πjk{{xj,→xJ},→xJ}{xi,→xI}{Xk,→xI} (3.33) p∑A=1B2A(X)i=1(n−1)!2Πjk{xi,→xI}{{xj,→xJ}{Xk,→xJ},→xI} (3.34) p∑A=1(TrBA)NiA=(−1)n(n−1)!Πik{{xk,→xI},→xI}

where

 (3.35) Πij=δij−1γ2(P2)ij

is the projection onto the normal space.

Proof.

Let us prove formula (3.32); the other formulas can be proven analogously. One rewrites

 (TrBA)BA(X)i =1(n−1)!2{xj,→xJ}{→xJ,njA}{xi,→xI}{→x