Riemannian Geometry and Multilinear Tensors with Vector Fields on Manifolds Md

Home , Affine connection, Differentiable manifold, Metric connection, Riemannian geometry

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.

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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 to an arbitrary manifold. Riemannian metrics, affine (e) ℝGiven 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 . Then the pair ( , ) ) is called a chart. 푖 푖 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, Dhaka-1230, Bangladesh,, PH: 880- 1724493092, e-mail- (b) g푝(푈, 푉) 0 푝 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 curve ( , ) is given by a set on (a) ( + ) = + 푥 푥 푥 smooth functions 푋 1 2 푋 1 푋 2 훾 ∶ 푎 푏 → 푀 푛 (b) 훻 푌 푌= 훻 +푌 훻 푌 = ( ) 1 2 1 2 (c) 훻푋 (+푋 푌) = 훻푋( 푌) +훻푋 푌 1 = 1 ( ) 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( ) and 푋 , ( ). ⎬ 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 ΓDefinition푥 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 푛 푑푥 휕 ( ) = ( ) + . 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 푛 푑푥 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 푗=1 푒푖 푗 훻 = [ ( ∑ ) 푋+훻 푒 ] tensor of rank (1,2). 푖 푖 훻 푇 푛 푛 푑푥 푗 푑푥 푗 푖 ∑푖=1 ∑푗=1 푑푡 푒푖푋 푒푗 푑푡 푋 훻푒 푒푗 = [ ( ) Proof. We need to prove that 푖 푗 푛 푛 푑푥 휕푋 휕 ( , ) = ( , ) = ( , ), ( ) and 푖=1 푗=1 푖 푗 ∑ ∑ 푑푡 휕푥 휕푥 훻 , ( ). 훻 훻 + ( ) ] 푖 푇 푓 푋 푌 푇 푋 푓 푌 푓 푇 푋 푌 ∀ 푓 ∈ ℱ 푀 푑푥 푗 푛 푘 휕 푘 By푋 the푌 ∈definition 픛 푀 of torsion, we get 푑푡 푘=1 푖푗 휕푥 푋 ∑ ⎾ 푥 = . + ( ) 푗 , , 푖 ( , ) = [ , ] 푑푋 휕 푑푥 휕 푛 푛 푗 푘 푗 푘 훻 ∑푗=1 푑푡 휕푥 ∑푖 푗 푘=1 푑푡 푋 ⎾푖푗 푥 휕푥 푓 푋 푌 ( ) ( ) = [ 푇 푓 푋 푌 = 훻 푌 − 훻( 푓)푋 − 푓 푋 푌 [ , ] 푘 푛 푑푋 훾̇ 푡 푘=1 푋 …푌 … … (1.01) ⇒ 훻 푋 푡 + ∑ 푑푡 (x)] . 푓 훻 푌 − 푌 푓 푋 − 푓 훻 푋 − 푓 푋 푌 , 푖 푛 푑푥 푗 푘 휕 푖 푗=1 푖푗 푘 Now take any g ( ), then ∑ 푑푡 푋 ⎾ 휕푥 Thus the equation for the parallel transport is [ , ]g = ∈ ℱ(푀g) ( (g)) + ( ) = 0 푘 , 푖 푓 푋 푌 = 푓 푋�(푌g) � −( 푌).푓 (푋g) (g) 푑푋 푛 푑푥 k ( ) = ( 푗 ) 푑푡 ∑푖 푗=1 푑푡 푋 Γi j 푥 푘 푘 � = 푓 푋푌 (g−) 푌 푓 (g푋) −( 푓 푌푋). (g) 0 0 This is a 푋system푡 of 푋-equations푖푛푖푡푖푎푙 for푐표푛푑푖푡푖표푛 -unknown functions ( ) with -initial conditions. A theorem from the theory푘 of = 푓 �[ 푋푌, ]g − 푌푋( ). �(g−) 푌 푓 푋 differential equations푛 says thatIJSER the solution푛 exists and which푋 푡is [ , ] = [ , ] ( ) . unique.푛 This completes the proof of this theorem. 푓 푋 푌 − 푌 푓 푋 (1.01) ∴Therefore,푓 푋 푌 equation푓 푋 푌 − 푌 becomes푓 푋

V. Torsion tensor and Riemann curvature tensor ( , ) = ( ) [ , ]

훻 In the mathematical field of differential geometry, the Riemann 푋 + (푌 ) 푇 푓 푋 푌 푓 훻 푌 − 푌 푓 푋 − 푓 훻 푋 − 푓 푋 푌 curvature tensor, or Riemannian–Christoffel tensor is the most standard way to express curvature of Riemannian manifolds. It = ( 푌[ 푓, 푋] ) associates a tensor to each point of a Riemannian manifold that 푋 푌 measures the extent to which the metric tensor is not locally = 푓 훻( 푌 , −). 훻 푋 − 푋 푌 isometric to a Euclidean space. Again, 훻 푓 푇 푋 푌 Definition 1.16 [5] Let be a smooth -dimensional ( , ) = ( , ) manifold, ( ) be the set of smooth functions and ( ) be the 훻 훻 vector space of smooth vector푀 fields. A tensor of rank푛 (1, ) on [ ( , ) = ( , )] 푇 푋 푓푌 − 푇 푓푌 푋 ℱ 푀 픛 푀 훻 훻 is a multi-linear map = ( , ) [as previous part] ∵ 푇 푋 푌 − 푇 푌 푋 퐴 푝 훻 ( ) × ( ) × … × ( ) ( ) 푀 = − 푓 푇( , 푌). 푋

훻 which satisfies 퐴 ∶ 픛 푀 픛 푀 픛 푀 → ℱ 푀 Therefore, is푓 a푇 tensor푋 푌of rank (1,2). , ,…, = , ,… , = 훻 Hence completes the proof. 푇 1 2 푝 1 2 푝 퐴 � 푓 푥 푥 푥 � = 퐴� 푥 , 푓푥,…, 푥 � ⋯ Definition 1.19 The curvature tensor , of an affine connection

+ ( ) ( ) ( ) × ( ) × ( ) ( ) 푋 푌 푌 훻 푓 훻 훻 푍 − 푌�푋 푓 �푍 − 푋 푓 훻 푍 ( ) [ , ] . 푅 ∶ 픛 푀 픛 푀 픛 푀 →( 픛, 푀, ) ( , , ) 훻 푋 푌 푋 − 푌 푓 훻 푍 − 푓 훻 훻 푍 − 푋 푌 푓 푍 where ( , , ) = 푋 푌 푍 ↦ 푅 [ 푋, ]푌. 푍 [ , ] 훻 푋 푌 푌 푋 푋 푌 −푓 훻 푋 푌 푍 Theorem푅 1.20푋 푌[6]푍 For all훻 affine훻 푍 − connection훻 훻 푍 −훻 , its푍 curvature = ( ) ( ) + ( is a tensor of rank (1, 3). 훻 푋 ) 푌 [ , 푌] 푋. 훻 푅 � 푋푌 푓 − 푌푋 푓 � 푍 [ 푓, 훻] 훻 푍−훻 훻 푍

Proof: We need to prove that 푋 푌 = [ , ] . + (−훻 , , )푍 −[ ,푋 ]푌.푓 푍 ( , , ) = ( , , ) 훻 = ( , , ). 훻 훻 푋 푌 푓 푍 푓 푅 푋 푌 푍 − 푋 푌 푓 푍 푅 푓 푋 푌 푍 =푅 ( 푋 ,푓 ,푌 푍 ) 훻 훻 Therefore,푓 푅 푋 푌 is푍 a tensor of rank (1, 3).

= 푅 (푋 ,푌,푓푍); ( ) and , , Hence completes훻 the proof. ( ). 훻 푅 By the definition of푓 curvature푅 푋 푌 tensor,푍 ∀ we푓 ∈ get ℱ 푀 푋 푌 푍 ∈ VI. Levi-Civita connections 픛 푀 ( , , ) Let be a smooth n-dimensional manifold, ( ) be the set of 훻 smooth functions, g be a smooth metric, ( ) be the vector space 푅= 푓 푋 푌 푍 [ , ] of smooth푀 vector fields and be an affine ℱconnection푀 on .

푓 푋 푌 푌 푓푋 푓 푋 푌 Then the covariant derivative on g with픛 푀 respect to is a = 훻 훻 푍 − 훻 훻( − 훻 ) 푍[ , ]– ( ) multilinear map, 훻 푀

푋 푌 푌 푋 푓 푋 푌 푌 푓 푋 훻 = 푓 훻 훻 푍 − 훻 ( 푓) 훻 푍 − 훻 푍[ , ] g ( ) × ( ) × ( ) ( ) + 푋 푌 푋 푌 푋 푓 (푋 푌) 푓 훻 훻 푍 − 푌 푓 훻 푍 − 푓 훻 훻 푍 − 훻 푍 훻 ∶ 픛 푀 픛( 푀, , 픛 ) 푀 →g( ℱ ,푀) 푌 푓 푋 = ( ) 훻 푍 푍 [ , ] where g( , )푍=푋 g푌( , ↦) 훻 g( 푋 푌, ) g( , ) + ( ) 푋 푌 푋 푌 푋 푋 푌 푍 푍 푍 푓 훻 훻 푍 − 푌 푓 훻 푍 − 푓 훻 훻 푍 − 푓 훻 푍 Definition훻 1.21푋 푌 [7] Let푍 M푋 be푌 a− smooth훻 푋 manifold푌 − 푋 equipped훻 푌 with a = ( ) 푋 [ , ] 푌 푓 훻 푍 smooth manifold metric g. There is a unique affine connection

푋 푌 푌 푋 푋 푌 g =푓 훻 (훻 ,푍 ,−).훻 훻 푍 − 훻 푍 on such that 훻 훻 Again, note that (a) is torsion free. 푓 푅 푋 푌 푍

(b) g = 0 This unique connection is called Levi-Civita ( , , ) = ( , ,IJSER) 훻 connection. 훻 훻 훻 Then푅 푋 푌 푍 − 푅 푌 푋 푍 Theorem 1.22 (The fundamental theorem of pseudo-Riemannian ( , g), ( , , ) = ( , , ) geometry) On a pseudo-Riemannian manifold there exists 훻 훻 a unique symmetric connection (Levi-Civita connection) which is

푅 푋 푓 푌 푍 = − 푅 푓(푌 ,푋 ,푍 ) [as previous part] compatible with the metric g. 푀 훻 = − 푓 푅( , 푌, 푋).푍 Proof: Let be a tangent vector to an arbitrary curve along which

Also, 훻 the vectors are parallel transported. Then we have, 푓 푅 푋 푌 푍 훾 0 = [g(X, Y)] = [( g)(X, Y)] + g( X, Y) ( , , ) i 훻 ∇ 훾 γ ∇ i +g(X, Y)∇i =푅 푋 푌 푓 푍 [ , ] = ( g) ). i 푋 푌 푌 푋 푋 푌 ∇ 훻 훻 푓 푍 − 훻 훻 푓 푍 − 훻 푓 푍 i 푗 푘 = ( ( ) + ( ( ). + ) 푖 푗푘 whereγ we푋 have푌 ∇ noted that X = Y = 0. Since this true for any 푋 푌 푌 푋 curves and vectors, we must have 훻 푌 푓 푍 푓 훻 푍 − 훻 푋 [푓 , 푍]( 푓).훻 푍 i i [ , ] ∇ ∇ 푋 푌 ( g) = 0. = ( ( ) ) + ( )− 푋 푌 ( 푓( )푍. 푓) 훻 푍

i jk 푋 푋 푌 푌 For metric tensor we know, ∇ 훻 푌 푓 (푍 ( )훻. ) 푓 훻 푍( − 훻) 푋[푓, 푍]( ).

푌 푌 푋 [ , ] −훻 푋 푓 푍 − 훻 푓 훻 푍 − 푋 푌 푓 푍 ( g) = g g g . x 푋 푌 i i ℓ jk ∂ jk ℓ j ik ℓ k ij = ( ( ) ) + ( ) + ( −푓) 훻 푍 ℓ Now∇ from the above,− Γ we can− write Γ as follows ∂ 푋 푋 푌 훻 푌 푓 푍 푌 푓 훻 푍 푋 푓 훻 푍 IJSER © 2014 http://www.ijser.org International Journal of Scientific & Engineering Research, Volume 5, Issue 9, September-2014 161 ISSN 2229-5518

Levi-Civita connection coefficients are = = and g g g = 0 … … … (1.02) x = . 휙 휙 −1 i i 푟휙 푟휆 ∂ jk ℓ j ik ℓ k ij Γ Γ 푟 ℓ − Γ − Γ 푟 ∂Now using cyclic permutations of ( , , ), we have 휙휙 ΓExample−푟 1.24 The induce map on is = + g g g = 0… … …(1.03) 2 x ℓ 푗 푘 . the non-vanishing components of the Levi- i i 2 푆 푔 푑휃⨂푑휃 ∂ j kℓ j k iℓ j ℓ ik Civita connections are − Γ − Γ 푠푖푛 휃 푑휙 ⨂ 푑휙 ∂ g g g = 0 … … … (1.04) x i i = ; = = . ∂ ℓj kℓ ij kj iℓ 휙 휙 k − Γ − Γ 휃 ∂The combination (1.02) + (1.03) + (1.04) yields Γ 휙휙 −푐표푠휃 푠푖푛휃 Γ 휃휙 Γ 휙휃 푐표푡휃 VII. Killing Vector Fields − g + g + g + g x x x A Killing vector field is a vector field on a Riemannian manifold ∂ ∂ ∂ i that preserves the metric. Killing fields are the infinitesimal − ℓ jk j kℓ k ℓj Τℓ j ik ∂ ∂ + g∂ 2 ( )g = 0… … …(1.05) generators of isometrics, that is, flows generated by Killing fields i i are continuous isometrics of the manifold. ℓ k ij j k iℓ where = 2 [Τ ] = − Γ and ( ) = + . The 1 Definition 1.25 [9] A vector field is a Killing vector field if the tensor i is antii -symmetrici iwith respecti to the i loweri indices Τℓ j Γ ℓ j Γℓ j − Γj ℓ Γ j k 2 �Γkj Γj k � Lie derivative with respect to of the metric vanishe i = ℓ j . 푋 i Τ i = 0. ℓ j j ℓ 푋 푔 ByΤ solving−Τ equation (1.05) for ( ), we have 푥 i In terms of the Levi-Civita connection,ℒ 푔 this is i 1 Γ j k = + + … … … (1.06) ( , ) + ( , ) = 0 ( ) j k 2 i i i 푌 푍 Γ j k � � �Τk j Τ j k� for all vectors and푔 훻 . 푋In푍 local푔 coordinates,푌 훻 푋 this amounts to the where are the christoffel symbols defined by Killing equation i 푌 푍 j k �1� + = 0. = + … … … (1.07) 2 x x x 푖 푖 푙 휕푔푘 푙 휕푔푗 푙 휕푔푗 푘 This condition is expressed훻푖푋푗 in훻 푗covariant푋푖 form. Therefore it is � � 푔 � j k − ℓ � Finally,푗 푘 the connection휕 coefficient휕 휕 is given by sufficient to establish it in a preferred coordinate system in order to have it hold in all coordinate systems. = ( ) + [ ] Γ i i i Examples 1.26 The vector field on a circle that points clockwise Γ j k Γ j k Γ j k and has the same length at each point is a Killing vector field, = + + + … … … (1.08) 푖 1 i i IJSERi since moving each point on the circle along this vector field

푗 푘 2 k j j k j k simply rotates the circle. The second� � term�Τ of theΤ lastΤ expression� of (1.08) is called contorsion, denoted by K : i VIII. Conformal Killing Vector Fields 1 jk = ( + + ) The term of the conformal Killing vector field is an extension of 2 푖 i i i the term of the Killing vector field. Conformal Killing vector 푗푘 k j j k j k So퐾 , Τ Τ Τ fields scale the Metric around a smooth function, while Killing vector fields do not scale the Metric. The conformal Killing = + vectors are the infinitesimal generators of conformal 푖 푖 푗 푘 푖 푗 푘 transformations. Γ � � 퐾 Now, 푗 (푘 ) = + is the another connection coefficient 푖 푖 푖 (1,2) Definition 1.27 [10] Let ( , ) be a Riemannian manifold and if is a tensor푗 푘 field푗 푘of type푗 푘 . Now we choose Γ Γ 푇 ( ). Then the vector field is a conformal Killing vector 푀 푔 푇 = so that field, if an infinitesimal displacement given by generates 푖 푖 a푋 conformal ∈ 픛 푀 transformation. 푋 푇 푗 푘 −퐾 푗 푘 휀 푋 =

푖 Example 1.28 Let be the coordinates of ( , ). The vector

푗 푘 푖 푘 푚 Γ � � = + . = 푗 푘 푥 휕 ℝ 훿 푙푘 휕푔푗 푙 휕푔푗 푘 푘 1 푖 푙 휕푔 is a conformal killing vector. 푘 푗 푘 푙 퐷 푥 휕푥 By construction,2 푔 � 휕푥 this is휕푥 symmetric− 휕푥 � and certainly unique given a metric. IX. Conclusion Example 1.23 Let metric on in polar coordinates is = The fundamental theorem of pseudo-Riemannian geometry is + . The non2 -vanishing components of the established using tensors on a manifold . In this theorem, I have 2 ℝ 푔 IJSER © 2014 푑푟 ⨂ 푑푟 푟 푑휙 ⨂ 푑휙 http://www.ijser.org 푀 International Journal of Scientific & Engineering Research, Volume 5, Issue 9, September-2014 162 ISSN 2229-5518 used metric connection which is the natural generalization of the connection defined in the classical geometry of surfaces. The covariantly constant metric∇ and vectors fields and , which are parallel transported along any curve are used in this theorem. 푔푖푗 푋 푌 In this paper, I have tried to set different types of examples and the proof of various theorems in elaborate way so that it can be helpful for further analysis.

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