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Intrinsic Differential Geometry with Geometric Calculus MM Research Preprints, 196{205 MMRC, AMSS, Academia Sinica No. 23, December 2004 Intrinsic Di®erential Geometry with Geometric Calculus Hongbo Li and Lina Cao Mathematics Mechanization Key Laboratory Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing 100080, China Abstract. Setting up a symbolic algebraic system is the ¯rst step in mathematics mechanization of any branch of mathematics. In this paper, we establish a compact symbolic algebraic framework for local geometric computing in intrinsic di®erential ge- ometry, by choosing only the Lie derivative and the covariant derivative as basic local di®erential operators. In this framework, not only geometric entities such as the curva- ture and torsion of an a±ne connection have elegant representations, but their involved local geometric computing can be simpli¯ed. Keywords: Intrinsic di®erential geometry, Cli®ord algebra, Mathematics mecha- nization, Symbolic geometric computing. 1. Introduction Mathematics mechanization focuses on solving mathematical problems with symbolic computation techniques, particularly on mathematical reasoning by algebraic manipulation of mathematical symbols. In di®erential geometry, the mechanization viz. mechanical the- orem proving, is initiated by [7] using local coordinate representation. On the other hand, in modern di®erential geometry the dominant algebraic framework is moving frames and di®erential forms [1], which are independent of local coordinates. For mechanical theorem proving in spatial surface theory, [3], [4], [5] proposed to use di®erential forms and moving frames as basic algebraic tools. It appears that di®erential geometry bene¯ts from both local coordinates and global invariants [6]. For extrinsic di®erential geometry, [2] proposed to use Cli®ord algebra and vector deriva- tive viz. Dirac operator as basic algebraic tools. This framework combines local and global representations, and conforms with the classical framework of vector analysis. The current paper intends to extend this representation to intrinsic di®erential geometry. In intrinsic di®erential geometry, there is already an algebraic system abundant in coordinate- free operators: Algebraic operators: tensor product ­, exterior product ^, interior product ix, contrac- tion C, the pairing c between a vector space and its dual, Riemannian metric tensor g, etc. Intrinsic Di®erential Geometry 197 Di®erential operators: exterior di®erentiation d, Lie derivative Lx, the bracket [x; y] of tangent vector ¯elds x; y, covariant di®erentiation Dx, curvature tensor R, torsion tensor T , etc. In our opinion, the existing algebraic system contains a lot of redundancy, which may complicate its mechanization. The ¯rst task should be to reduce the number of basic oper- ators in the above list, in order to gain both compact representation and e®ective symbolic manipulation for both local and global geometric computing. This task is ful¯lled in this paper. We propose a compact algebraic framework for local geometric computing in intrinsic di®erential geometry. We select the following operators as basic ones: Basic algebraic operators: tensor product ­, exterior product ^, interior product \¢". Basic local di®erential operators: Lie derivative La, vector derivative @, covariant dif- ferentiations ±a and ±. Basic global di®erential operators: Lie derivative Lx, exterior di®erentiation d, covari- ant di®erentiation ±. This framework is suitable for studying intrinsic properties of general vector bundles, a±ne connections, Riemannian vector bundles, etc, and can be extended to include spin structure and integration. Some typical bene¯ts of this framework include: 1. Exterior di®erentiation d is equal to the composition of the covariant di®erentiation ±a (or ± in the case of a torsion-free a±ne connection) and the wedge product. 2. The curvature and torsion tensors can be represented elegantly by covariant di®erenti- ation. By these representations, the famous Bianchi identities are given \transparent" algebraic representations. 3. Some geometric theorems can be given simpli¯ed proofs, e.g., Schur's Lemma in Rie- mannian geometry. This paper is organized as follows. In Section 2 we introduce some terminology and notations. In Section 3 we construct exterior di®erentiation locally by Lie derivative. In Section 4 we represent curvature tensor, torsion tensor and Bianchi identities locally by covariant di®erentiation. In Section 5 we represent the curvature tensor of Riemannian manifold explicitly as a 2-tensor of the tangent bivector bundle, and use it to give a simple proof of Schur's Lemma. 2. Some Terminology and Notations We use the same symbol \¢" to denote the pairing between a vector space and its dual space, the interior product of a tangent vector ¯eld and a di®erential form, and the inner product induced by a pseudo-Riemannian metric tensor. The explanation is that if no metric tensor occurs, then the dot symbol denotes the interior product which includes the pairing naturally; if there is a ¯xed pseudo-Riemannian metric tensor, then the tangent space and 198 H. Li and L. Cao cotangent space are identi¯ed, so are the pairing and the inner product induced by the metric. The latter identi¯cation is further justi¯ed by the fact that in Riemannian geometry, we use only the Levi-Civita covariant di®erentiation, which preserves both the inner product and the pairing. In this paper, we use the following extension [2] of the pairing \¢" from V and V¤ to their ¤ j ¤ Grassmann spaces ¤(V) and ¤(V ): for vectors xi 2 V and y 2 V , ½ 1 s 1 s (x1 ¢ (x2 ¢ (¢ ¢ ¢ (xr ¢ (y ^ ¢ ¢ ¢ ^ y ) ¢ ¢ ¢); if r · s; (x1 ^ ¢ ¢ ¢ ^ xr) ¢ (y ^ ¢ ¢ ¢ ^ y ) = 1 2 s (2.1) (¢ ¢ ¢ (x1 ^ ¢ ¢ ¢ ^ xr) ¢ y ) ¢ y ) ¢ ¢ ¢) ¢ y ); if r > s. The topic \vector derivative" is explored in detail in [2]. Let V and V¤ be an nD vector space and its dual space, Let fai j i = 1 : : : ng and fajg be a pair of dual bases of V and V¤ respectively. The vector derivative at a point x 2 V is the following V-valued ¯rst-order di®erential operator: ¯ Xn @ ¯ @ = ai ¯ : (2.2) x @a ¯ i=1 i x Here @=@aijx denotes the directional derivative at point x. Obviously, @ = @x is independent of the basis faig chosen, and for any vector a, Xn i a ¢ @ = @=@a; a = (a ¢ a )ai: (2.3) i=1 Henceforth we consider an nD di®erentiable manifold M. The tangent and cotangent bundles of M are denoted by TM and T ¤M respectively. The set of smooth tangent and cotangent vector ¯elds on M are denoted by ¡(TM) and ¡(T ¤M) respectively. The Lie derivatives of smooth scalar ¯eld f, tangent vector ¯eld y, and cotangent vector ¯eld ! with respect to a smooth tangent vector ¯eld x, are de¯ned by Lxf = x(f); Lxy = [x; y]; (2.4) y ¢ Lx(!) = Lx(y ¢ !) ¡ ! ¢ Lxy: Here [x; y] is the bracket de¯ned by [x; y](f) = x(y(f)) ¡ y(x(f)); 8 f 2 C1(M): (2.5) Below we introduce two notations. Let ¼ be a di®erential operator taking values in ¤(¡(T ¤M)). Then in ¼ ^ u ^ v, where u; v 2 ¤(¡(T ¤M)), the di®erentiation is carried out to the right hand side of ¼, i.e., to both u and v. If we want the di®erentiation be made only to u, we use ¼ ^ u_ ^ v to denote this, else if we want it be made only to v, we use ¼ ^ u· ^ v to denote this. If u itself is a di®erential operator taking values in ¤(¡(T ¤M)), then it has two parts: the ¤(¡(T ¤M))-part and the scalar-valued di®erential operator part. If the di®erentiation in ¼ is made only to the ¤(¡(T ¤M))-part of u, we use ¼ ^ u_ ^ v to denote this. If the di®erentiation is not made to the ¤(¡(T ¤M))-part of u, we use ¼ ^ u· ^ v to denote this. The scalar-valued Intrinsic Di®erential Geometry 199 di®erentiation parts s(¼), s(u) of ¼; u always act on v in the manner s(¼)s(u)v. If v is also a ¤(¡(T ¤M))-valued di®erential operator, and the di®erentiation in ¼ is made only to the ¤(¡(T ¤M))-part of v, we use ¼ ^ u ^ v_ or ¼ ^ u· ^ v_ to denote this, depending on how u is di®erentiated. 3. Exterior Di®erentiation De¯nition 3.1 A local natural basis faig of ¡(TM) at x 2 M is a set of n smooth tangent vector ¯elds de¯ned in a neighborhood N of x, such that there exists a local coordinate system fuig of N with the property @f 1 ai(f) = ; 8 f 2 C (N): (3.6) @ui i ¤ Let faig be a local basis of ¡(TM), and let fa g be its dual basis in ¡(T M). De¯ne Xn Xn a i i d = a ^ Lai = a Lai ^ : (3.7) i=1 i=1 Lemma 3.2 (1) Let faig be a local basis of ¡(TM). Then it is natural if only if [ai; aj] = 0 for any 1 · i < j · n. a 2 (2) (d ) = 0 if and only if faig is natural. Proof. We only prove the second part. By the de¯nition of da, to prove (da)2 = 0 we only need to prove (da)2f = 0 for any f 2 C1(M). Xn a 2 i j (d ) f = a ^ Lai (a aj(f)) i;j=1 Xn i j i j = (a ^ a ai(aj(f))) + aj(f)a ^ Lai (a ) i;j=1 X Xn i j i j k = [ai; aj](f)a ^ a ¡ aj(f)a ^ ([ai; ak] ¢ a )a i<j i;j;k=1 X Xn i j i k = [ai; aj](f)a ^ a ¡ [ai; ak](f)a ^ a i<jX i;k=1 i j = ¡ [ai; aj](f)a ^ a : i<j a 2 Thus (d ) f = 0 if and only if [ai; aj](f) = 0, i.e., faig is natural.
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