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Sum of Squares Manifolds: the Expressibility of the Laplace–Beltrami Operator on Pseudo-Riemannian Manifolds As a Sum of Squares of Vector Fields

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Sum of Squares Manifolds: the Expressibility of the Laplace–Beltrami Operator on Pseudo-Riemannian Manifolds As a Sum of Squares of Vector Fields TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 350, Number 10, October 1998, Pages 3943{3966 S 0002-9947(98)02016-9 SUM OF SQUARES MANIFOLDS: THE EXPRESSIBILITY OF THE LAPLACE{BELTRAMI OPERATOR ON PSEUDO-RIEMANNIAN MANIFOLDS AS A SUM OF SQUARES OF VECTOR FIELDS WILFRIED H. PAUS To my aunt Ingrid S. Keller Abstract. In this paper, we investigate under what circumstances the La- place–Beltrami operator on a pseudo-Riemannian manifold can be written as a sum of squares of vector fields, as is naturally the case in Euclidean space. We show that such an expression exists globally on one-dimensional man- ifolds and can be found at least locally on any analytic pseudo-Riemannian manifold of dimension greater than two. For two-dimensional manifolds this is possible if and only if the manifold is flat. These results are achieved by formulating the problem as an exterior dif- ferential system and applying the Cartan–K¨ahler theorem to it. 1. Introduction We study an n-dimensional real pseudo-Riemannian (sometimes called semi- Riemannian) manifold (that is, a pair ( ,g) of a differential manifold and its metric g) and a symmetric nondegenerateM (0,2)-tensor field on of constantM index. The central object of our investigation will be the Laplace–BeltramiM operator on such a manifold. 1.1. Notation and preliminaries. Arbitrary (open) subsets of a differential man- ifold are denoted by . The tangent space at p is always indicated by Tp ; M U ∈U U similarly we write Tp∗ for its dual. We also use the usual notation X( )forthe set of vector fields on U and write Ωr( ) for the set of differential r-formsU defined there. The neutral element;U in these andU other sets are denoted by 0 if the objects being considered are R-valued (e.g., numbers, functions) and by O if the objects have components (e.g., vector fields, matrices, differential forms). By the letter , we denote an open subset of the manifold which admits a local V n coordinate system x : R :q (x1(q),...,xn(q)). On such a set, we use V−→ 7→ Received by the editors December 30, 1996. 1991 Mathematics Subject Classification. Primary 58G03; Secondary 58A15, 53C21. Key words and phrases. Differential geometry, pseudo-Riemannian manifolds, the Laplace– Beltrami operator, exterior differential systems, Cartan–K¨ahler theory. This work was made possible through funding from the Australian Department of Employ- ment, Education and Training (OPRS), the Deutscher Akademischer Austauschdienst of Germany, the Australian Research Council Grant “Differential and Integral Operators”, and the Deutsche Forschungsgemeinschaft. c 1998 American Mathematical Society 3943 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 3944 WILFRIED H. PAUS the standard matrix representation of any pseudo-Riemannian metric g on with M respect to the canonical corresponding vector fields @ by writing @xi i ∂ ∂ G := (g ):= g , ij ∂x ∂x i j to obtain a matrix-valued mapping G from into the set of real invertible symmetric V 1 ij matrices. For the mapping from onto the inverse of G,wewriteG− =: (g ), n ij V i. e., j=1 g gjk = δik, the Kronecker delta. We use to denote the covariant derivatives on ( ,g). In order to perform covariantP differentiation∇ a certain level of regularity isM required, so we assume all vector fields and differential forms appearing in this paper to be at least once differentiable. We demand the same of the pseudo-Riemannian metric. Let us clarify the use of covariant differentiation. For any vector field X X( ) ∈ M the operator X denotes the pseudo-Riemannian connection acting on X( )as ∇ M usual (see [11], Chap. I, 9). Moreover, we consider X as a derivation on the Grassmann algebra of alternating§ differential forms with∇ the pointwise definition r (1.1) ( X σ)(Z ,...,Zr):=X(σ(Z ,...,Zr)) σ (Z ,..., XZi,...,Zr) ∇ 1 1 − 1 ∇ i=1 X r for all X, Z ,...,Zr X( )andanyr-form σ Ω ( ) (see [14], Chap. VIII, 1). 1 ∈ M ∈ M § In contrast to some references (e.g., [8]), we follow [25] by regarding X σ as an r-form when carrying out further covariant differentiation, i.e., for the second-order∇ covariant derivatives we set 2 r (1.2) XY σ = X Y σ XYσ for all X, Y X( )andσ Ω( ) ∇ ∇ ∇ −∇∇ ∈ M ∈ M with ( Y X σ)(Z1,...,Zr):=Y(( X σ)(Z1,...,Zr)) ∇ ∇ r∇ ( X σ)(Z ,..., Y Zj,...,Zr) − ∇ 1 ∇ i=1 X for all X, Y, Z1,...,Zr X( ), in conformity with (1.1). This is, however, just a matter of notation. ∈ M In the sequel we frequently utilise collections of vector fields X1,...,Xn X( ) defined on some subset of an n-dimensional differential manifold, which∈ formU U pointwise bases of the tangent spaces, i. e., where (Xi p)i is a basis of Tp for | U every p in . As is customary, we refer to such a list (Xi)i as a frame. Recall in this connectionU that a differential manifold is called parallelisable if and only if it admits a frame. The presence of a pseudo-Riemannian metric on a differential manifold leads to the notion of orthonormality on the tangent spaces. We recall a few definitions and facts in this connection (see, e.g., [17]). Definition 1.1. Let be an open subset of a pseudo-Riemannian manifold ( ,g) U M of dimension n. A list (Yi)i of n vector fields is, or the vector fields themselves are, called g-orthonormal if g(Yi,Yj)=jδij with j := g(Yj ,Yj)= 1 for all i, j =1,...,n. ± License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use SUM OF SQUARES MANIFOLDS 3945 The n-tuple (1,...,n) is usually called the signature of g (with respect to (Yi)i). The number of negative signs in a signature is the same for any choice of g-orthonormal vector fields, and is referred to as the index of the manifold. In the case of index 0 the pseudo-Riemannian metric g is positive definite and the prefix “pseudo” in all of the above is omitted. It is a well-known fact that one can construct locally a list of g-orthonormal vector fields on any pseudo-Riemannian manifold. In the Riemannian case this is even true globally, provided the manifold is parallelisable. Analogously we call a collection of differential 1-forms ω1,...,ωn on an open subset of a differential manifold a coframe if the list (ωk p)k formsabasisof U | T∗ for every p in . For every frame (Xi)i there exists a unique dual coframe pU U (ωk)k which is determined by the relation ωk(p)(Xi p)=δik for all i, k =1,...,n and every p in | U and vice versa. A coframe is called g-orthonormal if and only if its dual vector fields have this property. Remark 1.2. To express the property of g-orthonormality of (Yi) in terms of local coordinates (xi)i on some open subset ,write V⊆U n ∂ Yi = bij for all i =1,...,n. ∂xj j=1 X Then we have the following condition for the coefficients bij of these vector fields: n ∂ n ∂ n jδij = g (Yi,Yj)=g bik , bjl = bikbjlgkl. ∂xk ∂xl k l ! k;l X=1 X=1 X=1 Denoting B := (bij ), this identity reads, in matrix form, 1 J = B • G • B∗ or, equivalently, G− = B∗ • J • B, where we denote the standard matrix multiplication by • .HereJis the signa- 1 ture matrix associated with (Yi)i, i.e., J := (j δij )=J∗ =J− , which contains precisely as many minus signs on its main diagonal as the index of the manifold. 1.2. The Laplace{Beltrami operator. We deal with differential operators on a pseudo-Riemannian manifold ( ,g), in particular the Laplace–Beltrami operator M (1.3) ∆:=div grad . Here (1.4) g(grad f,X):=Xf and n (1.5) div X = ig ( Y X, Yi) , ∇ i i=1 X in either case for all X X( ), where (Yi)i is an arbitrary g-orthonormal frame ∈ M of signature (1,...,n) given locally on . With respect to local coordinates the Laplace–Beltrami operator (Laplacian forM short) is given by 1 n ∂ n ∂ (1.6) ∆= gij det G . ∂xj | |·∂xi det G j=1 i=1 ! | | X X p p License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 3946 WILFRIED H. PAUS Alternatively, one has n ij 2 ∆= g @ @ ∇ @xi @xj i;j=1 X with respect to covariant derivatives (cf. [8], 26). Recalling the definition of the n § Laplacian on R n ∂2 ∆ n = , R ∂x2 i=1 i X we may describe its structure by the following three properties: 2 n (i) ∆Rn is symmetric on the Lebesgue space L (R ) of square- integrable real-valued functions on Rn equipped with the usual 2 n 2 n inner product L (R ) L (R ) R :(f,g) n fg, (1.7) × → 7→ R (ii) ∆Rn can be written as a sum of squares of linear first-order differ- ential operators, and R (iii) these operators commute. Hence it is a natural question to ask whether the Laplacian on (and thus M on any (open) subset ) has similar features when we replace Rn by in the above. The first ofU⊆M these properties follows from the definition (1.3) ofM ∆ as a“div grad ” operator. Indeed, as in Rn, one has Green’s formula, which implies that the Laplace–Beltrami operator is symmetric (see [4] Theorem 3.18).
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