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Riemannian Geometry and Multilinear Tensors with Vector Fields on Manifolds Md

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Riemannian Geometry and Multilinear Tensors with Vector Fields on Manifolds Md International Journal of Scientific & Engineering Research, Volume 5, Issue 9, September-2014 157 ISSN 2229-5518 Riemannian Geometry and Multilinear Tensors with Vector Fields on Manifolds Md. Abdul Halim Sajal Saha Md Shafiqul Islam Abstract-In the paper some aspects of Riemannian manifolds, pseudo-Riemannian manifolds, Lorentz manifolds, Riemannian metrics, affine connections, parallel transport, curvature tensors, torsion tensors, killing vector fields, conformal killing vector fields are focused. The purpose of this paper is to develop the theory of manifolds equipped with Riemannian metric. I have developed some theorems on torsion and Riemannian curvature tensors using affine connection. A Theorem 1.20 named “Fundamental Theorem of Pseudo-Riemannian Geometry” has been established on Riemannian geometry using tensors with metric. The main tools used in the theorem of pseudo Riemannian are tensors fields defined on a Riemannian manifold. Keywords: Riemannian manifolds, pseudo-Riemannian manifolds, Lorentz manifolds, Riemannian metrics, affine connections, parallel transport, curvature tensors, torsion tensors, killing vector fields, conformal killing vector fields. —————————— —————————— I. Introduction (c) { } is a family of open sets which covers , that is, 푖 = . Riemannian manifold is a pair ( , g) consisting of smooth 푈 푀 manifold and Riemannian metric g. A manifold may carry a (d) ⋃ is푈 푖푖 a homeomorphism푀 from onto an open subset of 푀 ′ further structure if it is endowed with a metric tensor, which is a 푖 . 푖 푖 휑 푈 푈 natural generation푀 of the inner product between two vectors in 푛 ℝ to an arbitrary manifold. Riemannian metrics, affine (e) Given and such that , the map = connections,푛 parallel transport, curvature tensors, torsion tensors, ( ( ) killingℝ vector fields and conformal killing vector fields play from푖 푗 ) to 푖 푗 is infinitely푖푗 −1 푈 푈 푈 ∩ 푈 ≠ ∅ 휓 important role to develop the theorem of Riemannian manifolds. differentiable.푖 푗 푖 푗 푖 푖 푗 휑 휑푗 휑 푈 ∩ 푈 휑 푈 ∩ 푈 II. Riemannian manifolds Example 1.02 The Euclidean space is the most trivial example, where a single chart covers the whole푚 space and may A manifold is a topological space which locally looks like . be the identity map. ℝ Calculus on a manifold is assured by the the existence of smooth푛 휑 coordinate systems. Indeed, Riemannian manifold is ℝthe Definition 1.03 Let be a homeomorphism from an generalization of Riemannian metricIJSER with smooth manifold. 푛 open subset into 푖. Then푖 the pair ( , ) ) is called a chart. 휑푛 ∶ 푈 → ℝ 푖 푖 푖 Definition 1.04푈 [1]ℝ Let be a differentiable푈 휑 manifold. A IUBAT- International University of Business Agriculture and Technology, Riemannian metric g on is a type (0, 2) tensor field on Dhaka-1230, Bangladesh,, PH: 880- 1710226151, e-mail- which satisfies the following푀 axioms at each point [email protected] 푀 푀 (a) g ( , ) = g ( , ) 푝 ∈ 푀 IUBAT- International University of Business Agriculture and Technology, 푝 푝 Dhaka-1230, Bangladesh,, PH: 880- 1724493092, e-mail- (b) g (푈, 푉) 0 where푉 푈 the equality holds only when = 0. [email protected] 푝 Here 푈, 푈 ≥ and g = g| .In short g is a symmetric푈 IUBAT- International University of Business Agriculture and Technology, positive definite bilinear form and is a tangent space of Dhaka-1230, Bangladesh,, PH: 880- 1913004750, e-mail- 푝 푝 푝 푝 manifold푈 푉 at∈ a 푇 point푀 . [email protected] 푝 푇 푀 Definition푀 1.05 Let 푝be a differentiable manifold. A Riemannian metric g on is a pseudo-Riemannian metric if it satisfies the conditions ( ) and 푀( ) and if g ( , ) = 0 for any , = 0. 푀 then 푝 푝 Definition 1.01 is an n-dimensional differentiable manifold if 푖 푖푖 푈 푉 푈 ∈ 푇 푀 Definition푉 1.06 If g is Riemannian matric, all the eigenvalues are (a) is a topological푀 space, strictly positive and if g is pseudo-Riemannian some eigenvalues are negative. If there are positive and negative eigenvalues, (b) 푀 is provided with a family of pairs {( , )}, then the pair ( , ) is called the index of metric. If = , the metric is called a Lorentz metric. 푖 푗 푀 푈푖 휑푖 푖 푗 푖 푗 IJSER © 2014 http://www.ijser.org International Journal of Scientific & Engineering Research, Volume 5, Issue 9, September-2014 158 ISSN 2229-5518 Definition 1.07 Let ( , g) is Lorentzian. The elements of Definition 1.13 Let and be two tensor fields. Then the are divided into three classes as follows 푝 푀 푇 푀 covariant derivative along the field is defined as follows 1 2 (a) g( , ) > 0 is spacelike, 푇 푇 푋 ( 훻) = ( ) 푋+ ( ). (b) g(푈, 푈) = 0 → 푈 is lightlike, 푋 1 2 푋 1 2 1 푋 2 IV. Parallel훻 Transport푇 ⨂푇 훻 푇 ⨂푇 푇 ⨂ 훻 푇 (c) g(푈, 푈) < 0 → 푈 is timelike. Definition푈 푈1.08 [2]→ If 푈a smooth manifold admits a Riemannian Given a curve in a manifold , we may define the parallel metric g, the pair ( , g) is called a Riemannian manifold. If g is a transport of a vector along the curve. In geometry, parallel transport is a way of transporting푀 geometrical data along smooth pseudo-Riemannian metric, then ( , g) is푀 said to be a pseudo- curves in a manifold. If the manifold is equipped with an affine Riemannian manifold.푀 If g Lorentzian,( , g) is called a Lorentz connection, then this connection allows one to transport vectors manifold. 푀 푀 of the manifold along curves so that they stay parallel with Example 1.09 An -dimensional Euclidian space ( , ) is respect to the connection. Riemannian manifold and an -dimensional Minkowski푚 space ( , ) is a Lorentz푚 manifold. ℝ 훿 Definition 1.14 Let be a smooth -dimensional manifold 푚 푚 equipped with an affine connection . Let : ( , ) be a III.ℝ Affine휂 connection and covariant derivative smooth curve. A vector푀 field on ( ), 푛 ( ) is called parallel transport if the following equation is satisfied훻 훾 푎 푏 → 푀 A vector is a directional derivative acting on ( ) 훾 푡 푋 푡 as ( ). However, there is no directional derivative ( ) ( ) = 0, ( , ). 푋 ( , ) 푓 ∈ ℱ 푀 acting on a tensor field of type which arises naturally from 훾̇ 푡 the푋 differentiable ∶ 푓 → 푋 푓 structure of . What we need is an extra Here훻 푋( 푡) is the tangent∀ 푡vector∈ 푎 to푏 ( ) at the point . structure called the connection, which푝 푞 how tensor are transported along a curve. 푀 Theorem훾̇ 푡 1.15 [4] Let : ( , ) 훾 푡 be a smooth푡 curve on a manifold . For each ( , ) and for each ( ) , Definition 1.10 [3] Let be a smooth -dimensional manifold, 훾 푎 푏 → 푀 ( ) ( ) prove that there exist a unique0 vector field on0 훾 푡 0such ( ) be the set of smooth functions and ( ) be the vector that 푀 푡 ∈ 푎 푏 푋 ∈ 푇 푀 푀 푛 space of smooth vector fields. An affine connection on is a 푋 푡 훾 푡 ℱ 푀 픛 푀 (a) ( ) is parallel, map which is denoted by and defined by 푀 (a) 푋 (푡 ) = . 훻 ( ) × ( ) ( ) 0 0 Proof.푋 Let푡 ( , 푋 ) be a coordinate chart on a manifold at 훻 ∶ 픛 푀 ( 픛 ,푀 ) → 픛 푀 with coordinates ( , ,…, ). Then Such that 0 푋 IJSER 푈 휑 1 2 푛 푀 푡 푋 푌 ↦ 훻 푌 (a) A smooth curve푥 푥 (푥, ) is given by a set on (a) ( + ) = + smooth functions 푋 1 2 푋 1 푋 2 훾 ∶ 푎 푏 → 푀 푛 (b) 훻 푌 푌= 훻 +푌 훻 푌 = ( ) 푋1+푋2 푋1 푋2 1 1 (c) 훻 ( 푌) = 훻( 푌) +훻 푌 = ( ) 푥 2 :푥 2 푡 = ( ), 푋 푋 ⎫ (d) 훻 푓 푌= 푋 푓, 푌 푓 훻 푌 푥 :푥 푡 ⎪ 푖 푖 ⇒ 푥 푥 푡 푓 푋 푋 = ( ) ( ) and , ( ). ⎬ 훻 푌 푓 훻 푌 푛 푛 ⎪ where푥 푥 ( 푡, ⎭), = 1,2, … , Definition∀ 푓 ∈ ℱ 푀1.11 Let푋 ( 푌, ∈) 픛 be푀 a coordinate chart on a manifold with a coordinates ( , ,…, ). The functions ( ) are (a) 푡The ∈ 푎 vector푏 푖 ( ) is given푛 by 푈 휑1 2 푛 k called coordinate symbols of the affine connection . Here 푀 푥 푥 푥 Γi j 푥 ( ) is a function, where , = 1 , . ( ) = ( )푋 푡 for k 3 훻 푛 푖 휕 i j 푖 DefinitionΓ 푥 1.12푛 The vector field푖 푗 �� ��is푛� often called covariant 푖=1 휕푥 푋 푡 ∑ 푋= 푡 ( ) + ( ) + derivative of vector field ( ) along the vector field . It is 푋 to define the covariant derivative of훻 푓 by the ordinary directional 1 휕 2 휕 푋 푡 1 +푋 푡( ) 2 . derivative, 푓 ∈ ℱ 푀 푋 휕푥 휕푥 푓 푛 휕 ⋯ 푋 푡 푛 = ( ). Then ( ) is given by 휕푥 푋 For any , , it can be defined훻 푓 as follows푋 푓 ( ) =훾̇ 푡 푖 푑푥 휕 푛 ( ) = ( ) + . 푖=1 푖 푓 푌 훾̇ 푡 ∑ 푑푡 휕푥 훻푋 푓 푌 훻푋 푓 푌 푓 훻푋 푌 IJSER © 2014 http://www.ijser.org International Journal of Scientific & Engineering Research, Volume 5, Issue 9, September-2014 159 ISSN 2229-5518 = . + . + + . = , ,…, 1 2 푛 푑푥 휕 푑푥 휕 푑푥 휕 1 2 푛 1 2 푝 Now we have푑푡 ,휕푥 푑푡 휕푥 ⋯ 푑푡 휕푥 for any function 푓퐴(� 푥) and푥 ,푥 �, … , ( ). 푓 ∈ ℱ 푀 푥1 푥2 푥푝 ∈ 픛 푀 ( ) ( ) = Definition 1.17 A torsion of an affine connection , is a map 훻 푖 푛 푗 훾̇ 푡 푛 푑푥 푗=1 푗 훻 푋 푡 훻∑푖=1 푑푡 푒푖 ∑ 푋 푒 ( ) × ( ) (푇 ) 훻 = 훻 푖 푑푥 푛 푛 푗 푇 ∶ 픛 푀 픛 푀 → ( 픛, 푀) ( , ) 푖=1 푒푖 푗=1 푗 ∑ 푑푡 훻 � ∑ 푋 푒 � 훻 = ( 푖 푑푥 where ( , ) = 푋 푌 ↦ 푇 푋 [푌 , ]. 푛 푛 푗 푖=1 푗=1 푖 푗 훻 ∑ 푑푡 � ∑ +푒 푋 � 푒 ] 푋 푌 푇 푋 푌 훻 푌 − 훻 푋 − 푋 푌 푛 푗 Theorem 1.18 For all affine connection , its torson is a 푒푖 푗 = [ ( ∑푗=1) 푋+훻 푒 ] tensor of rank (1,2).
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