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Moving Frame Equations in Three Dimensional General Inner Product Space

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Moving Frame Equations in Three Dimensional General Inner Product Space Archive of SID Moving Frame Equations in Three Dimensional General Inner Product Space Ali Parsian Department of Mathematics, Tafresh University, Tafresh, Iran. [email protected] Abstract In this paper, we are going to generalize the Frenet-Serret formulas for the moving frames in three dimensional space 푅3, in the case that the space admits the general form of inner product. Keywords: Covariant derivative, Moving frame, Vector field. www.SID.ir Archive of SID Preliminaries Vectors are used widely in physics and engineering to describe forces, velocities, angular momentum, and many other concepts. To obtain a definition that is both practical and precise, we shall describe an “arrow” in 푅3 by giving its starting point 푝 and the change, or vector 푣, necessary to reach its end point 푝 + 푣. Strictly speaking, 푣 is just a point of 푅3. A tangent 3 3 vector 푣푝 to 푅 consists of two points of 푅 : its vector part 푣 and its point of application 푝. A vector field 푉 on 푅3 is a map that assigns to each point 푝 of 푅3 a tangent vector 푉(푝) to 푅3 at 푝. There is a natural algebra of vector fields. At each point 푝, the values 푉(푝) and 푊(푝) are 3 in the same vector space, the tangent space 푇푝푅 , consequently, the formula for the addition is thus the same as for addition of maps, (푉 + 푊)(푝) = 푉(푝) + 푊(푝) or all 푝 ∈ 푅3. If 푓 is a real-valued map on 푅3 and 푉 is a vector field on 푅3, then 푓푉 is defined to be the vector field on 푅3 such that (푓푉)(푝) = 푓(푝)푉(푝) for all 푝 ∈ 푅3. Let 푓 be a differentiable real-valued 3 3 [ ] 푑 ( ) map on 푅 , and let 푣푝be a tangent vector to 푅 , then the number 푣푝 푓 = 푑푡푓 푝 + 푡푣 |푡=0 is called the derivative of 푓 with respect to 푣푝. It can be seen that, if 푣푝 = (푣1, 푣2, 푣3)푝 is a 3 휕푓 tangent vector to 푅 , then 푣푝[푓] = ∑ 푣푖 (푝) (Marsden and Tromba, 2011). If also 푓 and 푔 휕푥푖 3 be maps on 푅 , 푣푝 and 푤푝 tangent vectors, 푎 and 푏 numbers, then (푎푣푝 + 푏푤푝)[푓] = 푎푣푝[푓] + 푏푤푝[푓], 푣푝[푎푓 + 푏푔] = 푎푣푝[푓] + 푏푣푝[푔], 푣푝[푓푔] = 푣푝[푓]푔(푝) + 푓(푝)푣푝[푔]. For a vector field 푉, 푉[푓] is the real-valued map whose value at each point 푝 is the number 푉(푝)[푓]. Similarly, if 푓, 푔, ℎ are real-valued maps, 푉, 푊 are vector fields on 푅3, and 푎, 푏 ∈ 푅, then (푓푉 + 푔푊)[ℎ] = (푓푉)[ℎ] + (푔푊)[ℎ], 푉[푎푓 + 푏푔] = 푎푉[푓] + 푏푉[푔], 푉[푓푔] = 푉[푓]푔 + 푓. 푉[푔]. Replacing 푓 by a vector field 푊 on 푅3 gives a vector field 푡 → 푊(푝 + 푡푣) on the curve 푡 → 푝 + 푡푣. Then the derivative of 푊 with respect to 푣 will be the derivative of 푡 → 푊(푝 + 푡푣) at 푡 = 0. In fact, if 푊 be a vector field on 푅3, and 푣 be a tangent vector field to 푅3 at the point 푝, then the covariant derivative of 푊 with respect to 푣 is the tangent vector ∇푣푊(푝) = 푑 ( ) ( ) 푑푡푊 푝 + 푡푣 |푡=0. Evidently ∇푣푝푊 measures the initial rate of change of 푊 푝 as 푝 moves in 3 the 푣 direction. If 푈1, 푈2, and 푈3 be the vector fields on (푅 , ≪, ≫) and 푊 = ∑ 푤푖푈푖 is a 3 vector field on (푅 , ≪, ≫), and 푣 is a tangent vector at 푝, then ∇푣푝푊 = ∑ 푣푝[푤푖]푈푖 (푝). Moreover, if 푣 and 푤 be tangent vectors to (푅3, ≪, ≫) at 푝, and let 푌 and 푍 be vector fields on (푅3, ≪, ≫), then for numbers 푎, 푏 and map 푓, ∇(푎푣+푏푤)푌(푝) = 푎∇푣푌(푝) + 푏∇푤푌(푝), ∇푣(푎푌 + 푏푍)(푝) = 푎∇푣푌(푝) + 푏∇푣푍(푝), ∇푣(푓푌)(푝) = 푣푝[푓]푌(푝) + 푓(푝)∇푣푌(푝), 푣푝[≪ 푌, 푍 ≫] =≪ ∇푣푌, 푍(푝) ≫ +≪ 푌(푝), ∇푣푍 ≫. Using the pointwise principle, we can take the covariant derivative of a vector field W with respect to a vector field 푉, rather than a single tangent vector 푣. The result is the vector field ∇푉푊 whose value at each point 푝 is ∇푉(푝)푊. It follows immediately from above considerations that if 푊 = ∑ 푤푖푈푖, then ∇푉푊 = ∑ 푉[푤푖]푈푖. If 푓, 푔 be differentiable maps, 푎, 푏 ∈ 푅 and 푉, 푊, 푌, and 푍 be vector fields (푅3, ≪, ≫), then as a result of the preceding identities we have, ∇(푓푉+푔푊)푌 = 푓∇푉푌 + 푔∇푊푌, www.SID.ir Archive of SID ∇푉(푎푌 + 푏푍) = 푎∇푉푌 + 푏∇푉푍, ∇푉(푓푌) = 푉[푓]푌 + 푓∇푉푌, 푉[≪ 푌, 푍 ≫] =≪ ∇푉푌, 푍 ≫ +≪ 푌, ∇푉푍 ≫. Introduction Frenet- Serret’s essential idea was very simple: To each point of a curve, he assigned a frame; then using orthonormal expansion expresses the rate of change of the frame in terms of the frame itself. This, of course, is just what the Frenet- Serret formulas do in the case of a curve (Parsian, 2016). In the next, we shall carry out this scheme for the Inner product space (푅3, ≪ , ≫). We shall see that geometry of curves and surfaces in 푅3 is not merely an analogue, but actually a corollary, of these basic results. Method of Moving Frames If 푉 and 푊 are vector fields on 푅3, then the inner product ≪ 푉, 푊 ≫ of 푉 and 푊 is the differentiable real-valued map on 푅 whose value at each point 푝 is ≪ 푉(푝), 푊(푝) ≫. The norm ||푉|| of 푉 is the real-valued map on 푅3 whose value at 푝 is ||푉(푝)||. Vector fields 3 3 퐸1, 퐸2, 퐸3 on (푅 , ≪, ≫) constitute a frame field on (푅 , ≪, ≫) provided where ≪ 퐸푖, 퐸푗 ≫= 훿푖푗(1 ≤ 푖, 푗 ≤ 3) where 훿푖푗 is the Kronecker delta. The following useful result is an immediate consequence of orthonormal expansion. 3 3 Lemma 2.1. Let 퐸1, 퐸2, 퐸3be a frame field on (푅 , ≪, ≫). If 푉 is a vector field on 푅 , then 푉 = ∑ 푓푖퐸푖, where the maps 푓푖 =≪ 푉, 퐸푖. If 푉 = ∑ 푓푖퐸푖 and 푉 = ∑ 푔푖퐸푖, then ≪ 푉, 푊 ≫= 1 2 2 ∑ 푓푖 ∙ 푔푖. In particular, ||푉|| = (∑ 푓푖 ) . The Covariant Derivatives of a Frame Field The Frenet- Serret formulas express the derivatives of T, N, B in terms of T, N, B, and thereby define curvature and torsion. We shall now do the same thing with an arbitrary frame field 3 퐸1, 퐸2, 퐸3 on (푅 , ≪, ≫), namely, express the covariant derivatives of these vector fields in terms of the vector fields themselves. We begin with the covariant derivative with respect to an arbitrary tangent vector 푣 at a point 푝. Then ∇푣퐸푖 = ∑ 푐푖푗(푣)퐸푗(푝) for 푖, 푗 = 1,2,3. By orthonormal expansion the coefficients of these equations are 푐푖푗(푣) =≪ ∇푣퐸푖, 퐸푗(푝) ≫. A 1- form 푓 on 푅3 is a real-valued map on the set of all tangent vectors to 푅3such that 푓 is linear at each point. 3 3 Lemma 3.1. Let 퐸1, 퐸2, 퐸3 be a frame field on (푅 , ≪, ≫) . For each tangent vector 푣 to 푅 at the point 푝, let 푐푖푗(푣) be defined as above. Then each 푐푖푗 is a 1-form, and 푐푖푗 + 푐푗푖 = 0. Proof: By definition, 푐푖푗 is a real-valued map on tangent vectors, so to verify that 푐푖푗 is a 1- form, it suffices to check the linearity condition. Using above considerations, we get 푐푖푗(푎푣 + 푏푤) =) =≪ ∇푎푣+푏푤퐸푖, 퐸푗(푝) ≫=≪ 푎∇푣퐸푖 + 푏∇푤퐸푖, 퐸푗(푝) ≫ = 푎 ≪ ∇푣퐸푖, 퐸푗(푝) ≫ +푏 ≪ ∇푤퐸푖, 퐸푗(푝) ≫= 푎푐푖푗(푣) + 푏 푐푖푗(푤). To prove that 푐푖푗 + 푐푗푖 = 0 we show that 푐푖푗(푣) + 푐푗푖(푣) = 0 for every tangent vector 푣. By definition of frame field, ≪ 퐸푖, 퐸푗 ≫= 훿푖푗, and since each Kronecker delta has constant value 0 or 1, then 푣[≪ 퐸푖, 퐸푗 ≫] = 0, so the above considerations yields ≪ ∇푣퐸푖, 퐸푗(푝) ≫ +≪ 퐸푗(푝), ∇푣퐸푗 ≫= 0 and the proof is complete. www.SID.ir Archive of SID The definition 푐푖푗(푣) =≪ ∇푣퐸푖, 퐸푗(푝) ≫ shows that 푐푖푗(푣) is the initial rate at which 퐸푖 rotates toward 퐸푗 as 푝 moves in the 푣 direction. Thus the 1-forms 푐푖푗contain this information for all tangent vectors to 푅3. 3 Theorem 3.2. For any vector field 푉 on (푅 , ≪, ≫), ∇푉퐸푖 = ∑ 푐푖푗(푉) 퐸푗(1 ≤ 푖, 푗 ≤ 3). 3 Proof: Let 푖 be fixed and 푝 ∈ 푅 , then according to the previous considerations ∇푉(푝)퐸푖 = ∑ 푐푖푗(푉(푝))퐸푗(푝), so ∇푉퐸푖 = ∑ 푐푖푗(푉) 퐸푗. When 푖 = 푗, the skew-symmetry condition 푐푖푗 + 푐푗푖 = 0 becomes 푐푖푖 = 0 for 푖 =1,2,3. Hence this condition has the effect of reducing the nine 1-forms 푐푖푗 for 1 ≤ 푖, 푗 ≤ 3 to essentially only three, say 푐12, 푐13, 푐23. Thus in expanded form, the equations in Theorem 3.2 become ∇푉퐸1 = 푐12(푉)퐸2 + 푐13(푉)퐸3, ∇푉퐸2 = −푐12(푉)퐸1 + 푐23(푉)퐸3, ∇푉퐸3 = − 푐13(푉)퐸1 − 푐23(푉)퐸2. These equations play a fundamental role in all the differential geometry of (푅3, ≪, ≫). The generalized Frenet -Serret formulas can be deduced from them, Theorem 3.3. Let 훽 be a unit-speed curve in (푅3, ≪, ≫) with 푘 > 0, and suppose that 3 퐸1, 퐸2, 퐸3 is a frame field on (푅 , ≪, ≫) such that the restriction of these vector fields to b gives the generalized Frenet- Serret frame field 푇, 푁, 퐵 of 훽. Then 푐12(푇) = 푘, 푐13(푇) = 0, 푐23(푇) = 휏.
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