# Riemannian Geometry and Multilinear Tensors with Vector

## Transcript Of Riemannian Geometry and Multilinear Tensors with Vector

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) {𝑈𝑖} is a family of open sets which covers 𝑀, that is,

Riemannian manifold is a pair (𝑀, g) consisting of smooth manifold 𝑀 and Riemannian metric g. A manifold may carry a further structure if it is endowed with a metric tensor, which is a

natural generation of the inner product between two vectors in

⋃𝑖 𝑈𝑖 = 𝑀.

(d) 𝜑𝑖 is a homeomorphism from 𝑈𝑖 onto an open subset 𝑈𝑖′ of ℝ𝑛.

IJSER ℝ𝑛 to an arbitrary manifold. Riemannian metrics, affine

connections, parallel transport, curvature tensors, torsion tensors, killing vector fields and conformal killing vector fields play important role to develop the theorem of Riemannian manifolds.

II. Riemannian manifolds

A manifold is a topological space which locally looks like ℝ𝑛.

(e) Given 𝑈𝑖 and 𝑈𝑗 such that 𝑈𝑖 ∩ 𝑈𝑗 ≠ ∅, the map 𝜓𝑖𝑗 = 𝜑𝑖 𝜑𝑗−1 from 𝜑𝑗(𝑈𝑖 ∩ 𝑈𝑗) to 𝜑𝑖( 𝑈𝑖 ∩ 𝑈𝑗) is inﬁnitely differentiable.

Example 1.02 The Euclidean space ℝ𝑚 is the most trivial example, where a single chart covers the whole space and 𝜑 may be the identity map.

Calculus on a manifold is assured by the the existence of smooth

coordinate systems. Indeed, Riemannian manifold is the generalization of Riemannian metric with smooth manifold.

Definition 1.03 Let 𝜑𝑖 ∶ 𝑈𝑖 → ℝ𝑛 be a homeomorphism from an open subset 𝑈𝑖 into ℝ𝑛. Then the pair (𝑈𝑖, 𝜑𝑖) ) is called a chart.

IUBAT- International University of Business Agriculture and Technology, Dhaka-1230, Bangladesh,, PH: 880- 1710226151, [email protected] IUBAT- International University of Business Agriculture and Technology, Dhaka-1230, Bangladesh,, PH: 880- 1724493092, [email protected]

IUBAT- International University of Business Agriculture and Technology, Dhaka-1230, Bangladesh,, PH: 880- 1913004750, [email protected]

Definition 1.01 𝑀 is an n-dimensional differentiable manifold if (a) 𝑀 is a topological space, (b) 𝑀 is provided with a family of pairs {(𝑈𝑖, 𝜑𝑖)},

Definition 1.04 [1] Let 𝑀 be a differentiable manifold. A Riemannian metric g on 𝑀 is a type (0, 2) tensor field on 𝑀 which satisfies the following axioms at each point 𝑝 ∈ 𝑀

(a) g𝑝(𝑈, 𝑉) = g𝑝( 𝑉, 𝑈)

(b) g𝑝(𝑈, 𝑈) ≥ 0 where the equality holds only when 𝑈 = 0.

Here 𝑈, 𝑉 ∈ 𝑇𝑝𝑀 and g𝑝 = g|𝑝.In short g𝑝 is a symmetric positive definite bilinear form and 𝑇𝑝𝑀 is a tangent space of manifold 𝑀 at a point 𝑝.

Definition 1.05 Let 𝑀 be a differentiable manifold. A Riemannian metric g on 𝑀 is a pseudo-Riemannian metric if it satisfies the conditions (𝑖) and (𝑖𝑖) and if g𝑝(𝑈, 𝑉) = 0 for any 𝑈 ∈ 𝑇𝑝𝑀, then 𝑉 = 0.

Definition 1.06 If g is Riemannian matric, all the eigenvalues are strictly positive and if g is pseudo-Riemannian some eigenvalues are negative. If there are 𝑖 positive and 𝑗 negative eigenvalues, then the pair (𝑖, 𝑗) is called the index of metric. If 𝑖 = 𝑗, the metric is called a Lorentz metric.

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Definition 1.07 Let (𝑀, g) is Lorentzian. The elements of 𝑇𝑝𝑀 are divided into three classes as follows

(a) g(𝑈, 𝑈) > 0 → 𝑈 is spacelike,

(b) g(𝑈, 𝑈) = 0 → 𝑈 is lightlike,

Definition 1.13 Let 𝑇1 and 𝑇2 be two tensor fields. Then the covariant derivative 𝛻𝑋 along the field 𝑋 is defined as follows

𝛻𝑋 (𝑇1⨂𝑇2) = (𝛻𝑋 𝑇1)⨂𝑇2 + 𝑇1⨂(𝛻𝑋 𝑇2).

(c) g(𝑈, 𝑈) < 0 → 𝑈 is timelike.

IV. Parallel Transport

Definition 1.08 [2] If a smooth manifold 𝑀 admits a Riemannian metric g, the pair (𝑀, g) is called a Riemannian manifold. If g is a pseudo-Riemannian metric, then (𝑀, g) is said to be a pseudoRiemannian manifold. If g Lorentzian, (𝑀, g) is called a Lorentz manifold.

Example 1.09 An 𝑚-dimensional Euclidian space (ℝ𝑚, 𝛿) is Riemannian manifold and an 𝑚-dimensional Minkowski space (ℝ𝑚, 𝜂) is a Lorentz manifold.

III. Affine connection and covariant derivative

Given a curve in a manifold 𝑀, we may define the parallel transport of a vector along the curve. In geometry, parallel transport is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection, then this connection allows one to transport vectors of the manifold along curves so that they stay parallel with respect to the connection.

Definition 1.14 Let 𝑀 be a smooth 𝑛-dimensional manifold equipped with an affine connection 𝛻. Let 𝛾: (𝑎, 𝑏) → 𝑀 be a smooth curve. A vector field on 𝛾(𝑡), 𝑋(𝑡) is called parallel transport if the following equation is satisfied

A vector 𝑋 is a directional derivative acting on 𝑓 ∈ ℱ(𝑀) as 𝑋 ∶ 𝑓 → 𝑋(𝑓). However, there is no directional derivative

𝛻𝛾̇ (𝑡) 𝑋(𝑡) = 0, ∀ 𝑡 ∈ (𝑎, 𝑏).

acting on a tensor field of type (𝑝, 𝑞) which arises naturally from

the differentiable structure of 𝑀. What we need is an extra

Here 𝛾̇(𝑡) is the tangent vector to 𝛾(𝑡) at the point 𝑡.

structure called the connection, which how tensor are transported

along a curve.

Theorem 1.15 [4] Let 𝛾: (𝑎, 𝑏) → 𝑀 be a smooth curve on a

IJSER Definition 1.10 [3] Let 𝑀 be a smooth 𝑛-dimensional manifold,

ℱ(𝑀) be the set of smooth functions and 𝔛(𝑀) be the vector space of smooth vector fields. An affine connection on 𝑀 is a map which is denoted by 𝛻 and defined by

𝛻 ∶ 𝔛(𝑀) × 𝔛(𝑀) → 𝔛(𝑀)

manifold 𝑀. For each 𝑡0 ∈ (𝑎, 𝑏) and for each 𝑋0 ∈ 𝑇𝛾(𝑡0)𝑀 , prove that there exist a unique vector field 𝑋(𝑡) on 𝛾(𝑡) such that

(a) 𝑋(𝑡) is parallel,

(a) 𝑋(𝑡0) = 𝑋0.

Such that ( 𝑋 , 𝑌 ) ↦ 𝛻𝑋 𝑌 (a) 𝛻𝑋 (𝑌1 + 𝑌2) = 𝛻𝑋𝑌1 + 𝛻𝑋𝑌2

Proof. Let (𝑈, 𝜑) be a coordinate chart on a manifold 𝑀 at 𝑡0 with coordinates ( 𝑥1, 𝑥2, … , 𝑥𝑛). Then

(a) A smooth curve 𝛾 ∶ (𝑎, 𝑏) → 𝑀 is given by a set on 𝑛 smooth functions

(b) 𝛻𝑋1+𝑋2 𝑌 = 𝛻𝑋1𝑌 + 𝛻𝑋2𝑌 (c) 𝛻𝑋 (𝑓 𝑌) = 𝑋(𝑓) 𝑌 + 𝑓 𝛻𝑋𝑌 (d) 𝛻𝑓 𝑋𝑌 = 𝑓 𝛻𝑋𝑌, ∀ 𝑓 ∈ ℱ(𝑀) and 𝑋 , 𝑌 ∈ 𝔛(𝑀).

𝑥1 = 𝑥1(𝑡)

𝑥2

=

𝑥2(𝑡)

⎫ ⎪

:

⇒ 𝑥𝑖 = 𝑥𝑖(𝑡),

:

⎬ ⎪

𝑥𝑛 = 𝑥𝑛(𝑡) ⎭

Definition 1.11 Let (𝑈, 𝜑) be a coordinate chart on a manifold 𝑀 with a coordinates (𝑥1, 𝑥2, … , 𝑥𝑛). The functions Γikj(𝑥) are

called coordinate symbols of the affine connection 𝛻. Here Γikj(𝑥) is a 𝑛3 function, where 𝑖, 𝑗 = �1��,�𝑛�.

where 𝑡 ∈ (𝑎, 𝑏), 𝑖 = 1,2, … , 𝑛 (a) The vector 𝑋(𝑡) is given by 𝑋(𝑡) = ∑𝑛𝑖=1 𝑋𝑖 (𝑡) 𝜕𝜕𝑥𝑖 for

Definition 1.12 The vector field 𝛻𝑋 𝑓 is often called covariant derivative of vector field 𝑓 ∈ ℱ(𝑀) along the vector field 𝑋. It is

to define the covariant derivative of 𝑓 by the ordinary directional derivative,

= 𝑋1(𝑡)

𝜕 + 𝑋2(𝑡) 𝜕 +

𝜕𝑥1

𝜕𝑥2

⋯ + 𝑋𝑛(𝑡) 𝜕 .

𝜕𝑥𝑛

𝛻𝑋 𝑓 = 𝑋(𝑓).

Then 𝛾̇(𝑡) is given by

For any 𝑓, 𝑌, it can be defined as follows 𝛻𝑋 (𝑓 𝑌) = (𝛻𝑋 𝑓)𝑌 + 𝑓 𝛻𝑋 𝑌.

𝛾̇(𝑡) = ∑𝑛𝑖=1 𝑑𝑑𝑥𝑡𝑖 𝜕𝜕𝑥𝑖

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= 𝑑𝑑𝑥𝑡1 . 𝜕𝜕𝑥1 + 𝑑𝑑𝑥𝑡2 . 𝜕𝜕𝑥2 + ⋯ + 𝑑𝑑𝑥𝑡𝑛 . 𝜕𝜕𝑥𝑛 .

= 𝑓𝐴� 𝑥1, 𝑥2, … , 𝑥𝑝�

Now we have,

for any function 𝑓 ∈ ℱ(𝑀) and 𝑥1, 𝑥2, … , 𝑥𝑝 ∈ 𝔛(𝑀).

𝛻𝛾̇ (𝑡) 𝑋(𝑡) = 𝛻 𝑛 𝑑𝑥𝑖 ∑𝑛𝑗=1 𝑋𝑗 𝑒𝑗

∑𝑖=1 𝑑𝑡 𝑒𝑖

= ∑𝑛𝑖=1 𝑑𝑥𝑖

𝑑𝑡

𝛻𝑒𝑖 � ∑𝑛𝑗=1 𝑋𝑗 𝑒𝑗 �

= ∑𝑛𝑖=1 𝑑𝑥𝑖 � ∑𝑛𝑗=1 𝑒𝑖 ( 𝑋𝑗� 𝑒𝑗

𝑑𝑡

+ ∑𝑛𝑗=1 𝑋𝑗 𝛻𝑒𝑖 𝑒𝑗 ] = ∑𝑛𝑖=1 ∑𝑛𝑗=1 [ ( 𝑑𝑑𝑥𝑡𝑖 𝑒𝑖𝑋𝑗)𝑒𝑗 + 𝑑𝑑𝑥𝑡𝑖 𝑋𝑗 𝛻𝑒𝑖 𝑒𝑗 ]

Definition 1.17 A torsion 𝑇𝛻 of an affine connection 𝛻, is a map 𝑇𝛻 ∶ 𝔛(𝑀) × 𝔛(𝑀) → 𝔛(𝑀)

(𝑋, 𝑌) ↦ 𝑇𝛻 (𝑋, 𝑌) where 𝑇𝛻 (𝑋 , 𝑌) = 𝛻𝑋 𝑌 − 𝛻𝑌 𝑋 − [ 𝑋, 𝑌 ]. Theorem 1.18 For all affine connection 𝛻 , its torson 𝑇𝛻 is a tensor of rank (1, 2).

= ∑𝑛𝑖=1 ∑𝑛𝑗=1 [ ( 𝑑𝑑𝑥𝑡𝑖 𝜕𝜕𝑋𝑥𝑖𝑗) 𝜕𝜕𝑥𝑗 + 𝑑𝑑𝑥𝑡𝑖 𝑋𝑗 ∑𝑛𝑘=1 ⎾𝑖𝑘𝑗 (𝑥) 𝜕𝜕𝑥𝑘 ]

= ∑𝑛𝑗=1 𝑑𝑑𝑋𝑡𝑗 . 𝜕𝜕𝑥𝑗 + ∑𝑛𝑖,𝑗,𝑘=1 𝑑𝑑𝑥𝑡𝑖 𝑋𝑗 ⎾𝑖𝑘𝑗 (𝑥) 𝜕𝜕𝑥𝑘

Proof. We need to prove that 𝑇𝛻 (𝑓 𝑋 , 𝑌) = 𝑇𝛻 ( 𝑋 , 𝑓 𝑌 ) = 𝑓 𝑇𝛻 (𝑋 , 𝑌), ∀ 𝑓 ∈ ℱ(𝑀) and 𝑋 , 𝑌 ∈ 𝔛(𝑀).

By the definition of torsion, we get

𝑇𝛻 (𝑓 𝑋 , 𝑌) = 𝛻𝑓 𝑋 𝑌 − 𝛻𝑌 𝑓 𝑋 − [ 𝑓 𝑋, 𝑌 ]

⇒ 𝛻𝛾̇ (𝑡) 𝑋(𝑡) = ∑𝑛𝑘=1 [ 𝑑𝑋𝑘

𝑑𝑡

= 𝑓 𝛻𝑋 𝑌 − 𝑌(𝑓)𝑋 − 𝑓 𝛻𝑌 𝑋 − [ 𝑓 𝑋, 𝑌 ]

IJSER + ∑𝑛𝑖,𝑗=1 𝑑𝑑𝑥𝑡𝑖 𝑋𝑗 ⎾𝑖𝑘𝑗(x)] 𝜕𝜕𝑥𝑘.

Thus the equation for the parallel transport is 𝑑𝑑𝑋𝑡𝑘 + ∑𝑛𝑖,𝑗=1 𝑑𝑑𝑥𝑡𝑖 𝑋𝑗 Γikj(𝑥) = 0 �

𝑋𝑘(𝑡0) = 𝑋𝑘0 (𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛)

… … … Now take any g ∈ ℱ(𝑀), then [ 𝑓 𝑋, 𝑌 ]g = 𝑓 𝑋� 𝑌(g)� − 𝑌(𝑓 𝑋(g))

= 𝑓 𝑋𝑌(g) − 𝑌(𝑓). 𝑋(g) − 𝑓 𝑌𝑋(g) = 𝑓 � 𝑋𝑌(g) − 𝑌𝑋(g)� − 𝑌(𝑓). 𝑋(g)

This is a system of 𝑛-equations for 𝑛-unknown functions 𝑋𝑘(𝑡) with 𝑛-initial conditions. A theorem from the theory of

= 𝑓 [𝑋, 𝑌]g − 𝑌(𝑓). 𝑋(g)

differential equations says that the solution exists and which is unique. This completes the proof of this theorem.

∴ [ 𝑓 𝑋, 𝑌 ] = 𝑓 [𝑋, 𝑌] − 𝑌(𝑓) 𝑋.

(1.01)

Therefore, equation (1.01) becomes

V. Torsion tensor and Riemann curvature tensor

𝑇𝛻 (𝑓 𝑋 , 𝑌) = 𝑓 𝛻𝑋 𝑌 − 𝑌(𝑓)𝑋 − 𝑓 𝛻𝑌 𝑋 − 𝑓 [𝑋, 𝑌]

In the mathematical field of differential geometry, the Riemann curvature tensor, or Riemannian–Christoffel tensor is the most standard way to express curvature of Riemannian manifolds. It associates a tensor to each point of a Riemannian manifold that measures the extent to which the metric tensor is not locally isometric to a Euclidean space.

Again,

+𝑌(𝑓) 𝑋 = 𝑓 ( 𝛻𝑋 𝑌 − 𝛻𝑌 𝑋 − [𝑋, 𝑌] ) = 𝑓 𝑇𝛻 (𝑋 , 𝑌).

Definition 1.16 [5] Let 𝑀 be a smooth 𝑛-dimensional manifold, ℱ(𝑀) be the set of smooth functions and 𝔛(𝑀) be the vector space of smooth vector fields. A tensor 𝐴 of rank (1, 𝑝) on 𝑀 is a multi-linear map

𝐴 ∶ 𝔛(𝑀) × 𝔛(𝑀) × … × 𝔛(𝑀) → ℱ(𝑀)

which satisfies

𝐴�𝑓 𝑥1, 𝑥2, … , 𝑥𝑝� = 𝐴� 𝑥1, 𝑓𝑥2, … , 𝑥𝑝� = ⋯

𝑇𝛻 ( 𝑋 , 𝑓𝑌) = − 𝑇𝛻 ( 𝑓𝑌 , 𝑋)

[ ∵ 𝑇𝛻 (𝑋 , 𝑌) = − 𝑇𝛻 (𝑌 , 𝑋)] = − 𝑓 𝑇𝛻 ( 𝑌 , 𝑋) [as previous part] = 𝑓 𝑇𝛻 (𝑋 , 𝑌).

Therefore, 𝑇𝛻 is a tensor of rank (1, 2). Hence completes the proof.

= 𝐴� 𝑥1, 𝑥2, … , 𝑓𝑥𝑝�

Definition 1.19 The curvature tensor 𝑅𝛻, of an affine connection 𝛻 , is a map

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𝑅𝛻 ∶ 𝔛(𝑀) × 𝔛(𝑀) × 𝔛(𝑀) → 𝔛(𝑀)

+𝑓 𝛻𝑋 𝛻𝑌 𝑍 − 𝑌�𝑋(𝑓)�𝑍 − 𝑋(𝑓)𝛻𝑌 𝑍

(𝑋, 𝑌, 𝑍) ↦ 𝑅𝛻 (𝑋, 𝑌, 𝑍)

− 𝑌(𝑓)𝛻𝑋 𝑍 − 𝑓 𝛻𝑌 𝛻𝑋 𝑍 − [𝑋, 𝑌]𝑓. 𝑍

where 𝑅𝛻 (𝑋, 𝑌, 𝑍) = 𝛻𝑋 𝛻𝑌 𝑍 − 𝛻𝑌 𝛻𝑋 𝑍 −𝛻[ 𝑋,𝑌 ]𝑍.

−𝑓 𝛻[ 𝑋,𝑌 ] 𝑍

Theorem 1.20 [6] For all affine connection 𝛻, its curvature 𝑅𝛻 is a tensor of rank (1, 3).

Proof: We need to prove that

𝑅𝛻 (𝑓 𝑋 , 𝑌, 𝑍) = 𝑅𝛻 ( 𝑋 , 𝑓 𝑌, 𝑍 ) = 𝑅𝛻 ( 𝑋 , 𝑌, 𝑓𝑍 ) = 𝑓 𝑅𝛻 (𝑋 , 𝑌, 𝑍); ∀ 𝑓 ∈ ℱ(𝑀) and 𝑋 , 𝑌, 𝑍 ∈

𝔛(𝑀). By the definition of curvature tensor, we get

𝑅𝛻 (𝑓 𝑋, 𝑌, 𝑍)

= 𝛻𝑓 𝑋 𝛻𝑌 𝑍 − 𝛻𝑌 𝛻𝑓𝑋 − 𝛻[ 𝑓 𝑋,𝑌 ] 𝑍

= 𝑓 𝛻𝑋 𝛻𝑌 𝑍 − 𝛻𝑌 ( 𝑓 𝛻𝑋 𝑍 ) − 𝛻𝑓[ 𝑋,𝑌 ]–𝑌(𝑓)𝑋 𝑍

= �𝑋𝑌(𝑓) − 𝑌𝑋(𝑓)�𝑍 + 𝑓(𝛻𝑋 𝛻𝑌 𝑍−𝛻𝑌 𝛻𝑋 𝑍 −𝛻[ 𝑋,𝑌 ] 𝑍) − [𝑋, 𝑌]𝑓. 𝑍

= [𝑋, 𝑌]𝑓. 𝑍 + 𝑓 𝑅𝛻 (𝑋 , 𝑌, 𝑍) − [𝑋, 𝑌]𝑓. 𝑍

= 𝑓 𝑅𝛻 (𝑋 , 𝑌, 𝑍).

Therefore, 𝑅𝛻 is a tensor of rank (1, 3). Hence completes the proof.

VI. Levi-Civita connections

Let 𝑀 be a smooth n-dimensional manifold, ℱ(𝑀) be the set of smooth functions, g be a smooth metric, 𝔛(𝑀) be the vector space of smooth vector fields and 𝛻 be an affine connection on 𝑀. Then the covariant derivative on g with respect to 𝛻 is a multilinear map,

= 𝑓 𝛻𝑋 𝛻𝑌 𝑍 − 𝑌(𝑓) 𝛻𝑋 𝑍 − 𝑓 𝛻𝑌 𝛻𝑋 𝑍 − 𝛻𝑓[ 𝑋,𝑌 ] 𝑍

𝛻g ∶ 𝔛(𝑀) × 𝔛(𝑀) × 𝔛(𝑀) → ℱ(𝑀)

IJSER + 𝛻𝑌(𝑓)𝑋𝑍

= 𝑓 𝛻𝑋 𝛻𝑌 𝑍 − 𝑌(𝑓) 𝛻𝑋 𝑍 − 𝑓 𝛻𝑌 𝛻𝑋 𝑍 − 𝑓 𝛻[ 𝑋,𝑌 ] 𝑍 +𝑌(𝑓) 𝛻𝑋 𝑍

= 𝑓 ( 𝛻𝑋 𝛻𝑌 𝑍 − 𝛻𝑌 𝛻𝑋 𝑍 − 𝛻[ 𝑋,𝑌 ] 𝑍 )

= 𝑓 𝑅𝛻 (𝑋 , 𝑌, 𝑍).

Again, note that

( 𝑍, 𝑋 , 𝑌 ) ↦ 𝛻𝑍g(𝑋, 𝑌) where 𝛻𝑍g(𝑋, 𝑌) = 𝑍g(𝑋, 𝑌) − g(𝛻𝑍𝑋, 𝑌) − g(𝑋, 𝛻𝑍𝑌)

Definition 1.21 [7] Let M be a smooth manifold equipped with a smooth manifold metric g. There is a unique affine connection 𝛻 on g such that

(a) 𝛻 is torsion free.

𝑅𝛻 ( 𝑋, 𝑌, 𝑍) = − 𝑅𝛻 ( 𝑌, 𝑋, 𝑍) Then

𝑅𝛻 (𝑋, 𝑓 𝑌, 𝑍) = − 𝑅𝛻 (𝑓 𝑌, 𝑋, 𝑍) = − 𝑓 𝑅 𝛻 (𝑌, 𝑋, 𝑍) [as previous part]

(b) 𝛻g = 0 This unique connection is called Levi-Civita connection.

Theorem 1.22 (The fundamental theorem of pseudo-Riemannian geometry) On a pseudo-Riemannian manifold (𝑀, g), there exists a unique symmetric connection (Levi-Civita connection) which is compatible with the metric g.

Also,

= 𝑓 𝑅𝛻 (𝑋 , 𝑌, 𝑍).

Proof: Let 𝛾 be a tangent vector to an arbitrary curve along which the vectors are parallel transported. Then we have,

𝑅𝛻 (𝑋, 𝑌, 𝑓 𝑍) = 𝛻𝑋 𝛻𝑌 𝑓 𝑍 − 𝛻𝑌 𝛻𝑋 𝑓 𝑍 − 𝛻[ 𝑋,𝑌 ] 𝑓 𝑍

= 𝛻𝑋 (𝑌(𝑓)𝑍 + 𝑓 𝛻𝑌 𝑍 − 𝛻𝑌 ( 𝑋(𝑓). 𝑍 + 𝑓 𝛻𝑋 𝑍) −[𝑋, 𝑌](𝑓). 𝑍 𝑓 𝛻[ 𝑋,𝑌 ] 𝑍

= 𝛻𝑋 ( 𝑌(𝑓)𝑍 ) + 𝛻𝑋 (𝑓 𝛻𝑌 𝑍 ) − 𝛻𝑌 (𝑋(𝑓). 𝑍) −𝛻𝑌 ( 𝑋(𝑓). 𝑍)− 𝛻𝑌 (𝑓 𝛻𝑋 𝑍) − [𝑋, 𝑌](𝑓). 𝑍 −𝑓 𝛻[ 𝑋,𝑌 ] 𝑍

= 𝛻𝑋 ( 𝑌(𝑓)𝑍 ) + 𝑌(𝑓) 𝛻𝑋 𝑍 + 𝑋(𝑓)𝛻𝑌 𝑍

0 = ∇𝛾[g(X, Y)] = γi[(∇i g)(X, Y)] + g(∇iX, Y) +g(X, ∇iY)

= γi 𝑋𝑗𝑌𝑘(∇𝑖g)𝑗𝑘).

where we have noted that ∇i X = ∇i Y = 0. Since this true for any curves and vectors, we must have

(∇ig)jk = 0.

For metric tensor we know,

(∇ℓg)jk = ∂∂xℓ gjk − Γℓi jgik − Γℓi kgij. Now from the above, we can write as follows

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∂∂xℓ gjk − Γℓi jgik − Γℓi kgij = 0 … … … (1.02)

Levi-Civita

connection

coefficients

are

Γ

𝜙 𝑟𝜙

=

Γ

𝜙 𝑟𝜆

=

𝑟−1

and

Γ

𝑟 𝜙𝜙

= −𝑟.

Now using cyclic permutations of (ℓ, 𝑗, 𝑘), we have ∂∂xj gkℓ − Γij kgiℓ − Γij ℓ gik = 0 … … … (1.03)

∂∂xk gℓj − Γikℓ gij − Γikjgiℓ = 0 … … … (1.04)

Example 1.24 The induce map on 𝑆2 is 𝑔 = 𝑑𝜃⨂𝑑𝜃 + 𝑠𝑖𝑛2𝜃 𝑑𝜙 ⨂ 𝑑𝜙. the non-vanishing components of the LeviCivita connections are

Γ

𝜃 𝜙𝜙

= −𝑐𝑜𝑠𝜃 𝑠𝑖𝑛𝜃 ;

Γ

𝜙 𝜃𝜙

=

Γ

𝜙 𝜙𝜃

= 𝑐𝑜𝑡𝜃.

The combination −(1.02) + (1.03) + (1.04) yields

− ∂∂xℓ gjk + ∂∂xj gkℓ + ∂∂xk gℓj + Τℓi jgik +Τiℓ kgij − 2Γi(j k)giℓ = 0 … … … (1.05)

where Τℓi j = 2Γ[iℓ j] = Γℓi j − Γij ℓ and Γi(j k) = 12 �Γki j + Γji k �. The

tensor

Τ

i ℓ

j

is

anti-symmetric

with

respect

to

the

lower

indices

Τℓi j = −Τij ℓ.

By solving equation (1.05) for Γi(j k), we have

VII. Killing Vector Fields

A Killing vector field is a vector field on a Riemannian manifold that preserves the metric. Killing fields are the infinitesimal generators of isometrics, that is, flows generated by Killing fields are continuous isometrics of the manifold.

Definition 1.25 [9] A vector field 𝑋 is a Killing vector field if the Lie derivative with respect to 𝑋 of the metric 𝑔 vanishe

ℒ𝑥𝑔 = 0.

In terms of the Levi-Civita connection, this is

Γ

i (j

k

)

=

�

i

�

+

1

�Τ

i k

j

+

Τ

i j

k

�

…

…

…

(1.06)

jk 2

𝑔( 𝛻𝑌 𝑋, 𝑍) + 𝑔(𝑌, 𝛻𝑍 𝑋) = 0

IJSER where �jik� are the christoffel symbols defined by

�𝑗𝑖𝑘� = 12 𝑔𝑖 𝑙 �𝜕𝜕𝑔x𝑘j𝑙 + 𝜕𝜕𝑔x𝑗k𝑙 − 𝜕𝜕𝑔x𝑗ℓ𝑘� … … … (1.07)

Finally, the connection coefficient Γ is given by

Γ

i j

k

=

Γ

i (j

k)

+

Γ

i [j

k]

for all vectors 𝑌 and 𝑍. In local coordinates, this amounts to the Killing equation

𝛻𝑖𝑋𝑗 + 𝛻𝑗𝑋𝑖 = 0.

This condition is expressed in covariant form. Therefore it is sufficient to establish it in a preferred coordinate system in order to have it hold in all coordinate systems.

Examples 1.26 The vector field on a circle that points clockwise

=

�𝑗𝑖𝑘�

+

1 2

�Τki

j + Τji

k + Τij k�

…

…

… (1.08)

and has the same length at each point is a Killing vector field, since moving each point on the circle along this vector field

The second term of the last expression of (1.08) is called contorsion, denoted by Ki jk:

simply rotates the circle.

VIII. Conformal Killing Vector Fields

𝐾𝑖 = 1 (Τi + Τi + Τi )

𝑗𝑘 2 k j j k

j k

The term of the conformal Killing vector field is an extension of the term of the Killing vector field. Conformal Killing vector

So,

Γ𝑗𝑖𝑘

=�

𝑖

�

+

𝐾

𝑖 𝑗

𝑘

𝑗 𝑘

Now,

Γ

𝑖 (𝑗

𝑘)

=

Γ

𝑖 𝑗

𝑘

+

𝑇

𝑖 𝑗

𝑘

is the another connection coefficient

if 𝑇 is a tensor field of type (1,2). Now we choose

𝑇

𝑖 𝑗

𝑘

=

−𝐾

𝑖 𝑗

𝑘

so that

Γ

𝑖 𝑗

𝑘

=�

𝑖

�

𝑗 𝑘

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 transformations.

Definition 1.27 [10] Let (𝑀, 𝑔) be a Riemannian manifold and 𝑋 ∈ 𝔛(𝑀). Then the vector field 𝑋 is a conformal Killing vector field, if an infinitesimal displacement given by 𝜀 𝑋 generates a conformal transformation.

Example 1.28 Let 𝑥𝑘 be the coordinates of (ℝ𝑚, 𝛿). The vector

= 12 𝑔𝑖 𝑙 �𝜕𝜕𝑔𝑥𝑙𝑗𝑘 + 𝜕𝜕𝑔𝑥𝑗𝑘𝑙 − 𝜕𝜕𝑔𝑥𝑗𝑙𝑘�. By construction, this is symmetric and certainly unique given a

𝐷 = 𝑥𝑘 𝜕𝜕𝑥𝑘 is a conformal killing vector.

metric.

IX. Conclusion

Example 1.23 Let metric on ℝ2 in polar coordinates is 𝑔 = 𝑑𝑟 ⨂ 𝑑𝑟 + 𝑟2𝑑𝜙 ⨂ 𝑑𝜙. The non-vanishing components of the

The fundamental theorem of pseudo-Riemannian geometry is established using tensors on a manifold 𝑀. In this theorem, I have

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used metric connection ∇ which is the natural generalization of

the connection defined in the classical geometry of surfaces. The

covariantly constant metric 𝑔𝑖𝑗 and vectors fields 𝑋 and 𝑌, which

are parallel transported along any curve are used in this theorem.

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.

References

[1] Nakahara, M. 1990. Geometry, Topology and Physics, Shizuoka, Japan.

[2] Auslander, L. and Mackenzie, R. E. 1963. Introduction to differential manifolds, McGra Hill, New York.

[3] Ahmed, K. M. 2003. Curvature of Riemannian Manifolds, The Dhaka University Journal of Science, 25-29.

[4] Islam, J. Chrish, 1989. Modern Differential Geometry for Physicists, World Scientific Publishing Co. Pte. Ltd.

[5] Ahmed, K. M. 2003. A note on affine connection on Differetial Manifolds, The Dhaka University Journal of

IJSER Science, 63-68.

[6] Choquer-Bruhat, Y. and De Witt-Morette, C. DillardBleick, M. 1982. Analysis Manifolds and Physics edition, Amterdam, North-Holland.

[7] Brickell, F. and Clark, R. S. 1970. Differential Manifolds, Van Nostrand Reinhold Company, London

[8] Bott, R. and Tu, L. W. 1982. Different forms in Algebric Topology, New York.

[9] Chevally, C. 1956 Fundamental concepts of Algebra, Academic Press, New York.

[10] Donson, C. T. J. and Poston, T. 1977. Tensor Geometry, Pitman, London,.

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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) {𝑈𝑖} is a family of open sets which covers 𝑀, that is,

Riemannian manifold is a pair (𝑀, g) consisting of smooth manifold 𝑀 and Riemannian metric g. A manifold may carry a further structure if it is endowed with a metric tensor, which is a

natural generation of the inner product between two vectors in

⋃𝑖 𝑈𝑖 = 𝑀.

(d) 𝜑𝑖 is a homeomorphism from 𝑈𝑖 onto an open subset 𝑈𝑖′ of ℝ𝑛.

IJSER ℝ𝑛 to an arbitrary manifold. Riemannian metrics, affine

connections, parallel transport, curvature tensors, torsion tensors, killing vector fields and conformal killing vector fields play important role to develop the theorem of Riemannian manifolds.

II. Riemannian manifolds

A manifold is a topological space which locally looks like ℝ𝑛.

(e) Given 𝑈𝑖 and 𝑈𝑗 such that 𝑈𝑖 ∩ 𝑈𝑗 ≠ ∅, the map 𝜓𝑖𝑗 = 𝜑𝑖 𝜑𝑗−1 from 𝜑𝑗(𝑈𝑖 ∩ 𝑈𝑗) to 𝜑𝑖( 𝑈𝑖 ∩ 𝑈𝑗) is inﬁnitely differentiable.

Example 1.02 The Euclidean space ℝ𝑚 is the most trivial example, where a single chart covers the whole space and 𝜑 may be the identity map.

Calculus on a manifold is assured by the the existence of smooth

coordinate systems. Indeed, Riemannian manifold is the generalization of Riemannian metric with smooth manifold.

Definition 1.03 Let 𝜑𝑖 ∶ 𝑈𝑖 → ℝ𝑛 be a homeomorphism from an open subset 𝑈𝑖 into ℝ𝑛. Then the pair (𝑈𝑖, 𝜑𝑖) ) is called a chart.

IUBAT- International University of Business Agriculture and Technology, Dhaka-1230, Bangladesh,, PH: 880- 1710226151, [email protected] IUBAT- International University of Business Agriculture and Technology, Dhaka-1230, Bangladesh,, PH: 880- 1724493092, [email protected]

IUBAT- International University of Business Agriculture and Technology, Dhaka-1230, Bangladesh,, PH: 880- 1913004750, [email protected]

Definition 1.01 𝑀 is an n-dimensional differentiable manifold if (a) 𝑀 is a topological space, (b) 𝑀 is provided with a family of pairs {(𝑈𝑖, 𝜑𝑖)},

Definition 1.04 [1] Let 𝑀 be a differentiable manifold. A Riemannian metric g on 𝑀 is a type (0, 2) tensor field on 𝑀 which satisfies the following axioms at each point 𝑝 ∈ 𝑀

(a) g𝑝(𝑈, 𝑉) = g𝑝( 𝑉, 𝑈)

(b) g𝑝(𝑈, 𝑈) ≥ 0 where the equality holds only when 𝑈 = 0.

Here 𝑈, 𝑉 ∈ 𝑇𝑝𝑀 and g𝑝 = g|𝑝.In short g𝑝 is a symmetric positive definite bilinear form and 𝑇𝑝𝑀 is a tangent space of manifold 𝑀 at a point 𝑝.

Definition 1.05 Let 𝑀 be a differentiable manifold. A Riemannian metric g on 𝑀 is a pseudo-Riemannian metric if it satisfies the conditions (𝑖) and (𝑖𝑖) and if g𝑝(𝑈, 𝑉) = 0 for any 𝑈 ∈ 𝑇𝑝𝑀, then 𝑉 = 0.

Definition 1.06 If g is Riemannian matric, all the eigenvalues are strictly positive and if g is pseudo-Riemannian some eigenvalues are negative. If there are 𝑖 positive and 𝑗 negative eigenvalues, then the pair (𝑖, 𝑗) is called the index of metric. If 𝑖 = 𝑗, the metric is called a Lorentz metric.

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Definition 1.07 Let (𝑀, g) is Lorentzian. The elements of 𝑇𝑝𝑀 are divided into three classes as follows

(a) g(𝑈, 𝑈) > 0 → 𝑈 is spacelike,

(b) g(𝑈, 𝑈) = 0 → 𝑈 is lightlike,

Definition 1.13 Let 𝑇1 and 𝑇2 be two tensor fields. Then the covariant derivative 𝛻𝑋 along the field 𝑋 is defined as follows

𝛻𝑋 (𝑇1⨂𝑇2) = (𝛻𝑋 𝑇1)⨂𝑇2 + 𝑇1⨂(𝛻𝑋 𝑇2).

(c) g(𝑈, 𝑈) < 0 → 𝑈 is timelike.

IV. Parallel Transport

Definition 1.08 [2] If a smooth manifold 𝑀 admits a Riemannian metric g, the pair (𝑀, g) is called a Riemannian manifold. If g is a pseudo-Riemannian metric, then (𝑀, g) is said to be a pseudoRiemannian manifold. If g Lorentzian, (𝑀, g) is called a Lorentz manifold.

Example 1.09 An 𝑚-dimensional Euclidian space (ℝ𝑚, 𝛿) is Riemannian manifold and an 𝑚-dimensional Minkowski space (ℝ𝑚, 𝜂) is a Lorentz manifold.

III. Affine connection and covariant derivative

Given a curve in a manifold 𝑀, we may define the parallel transport of a vector along the curve. In geometry, parallel transport is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection, then this connection allows one to transport vectors of the manifold along curves so that they stay parallel with respect to the connection.

Definition 1.14 Let 𝑀 be a smooth 𝑛-dimensional manifold equipped with an affine connection 𝛻. Let 𝛾: (𝑎, 𝑏) → 𝑀 be a smooth curve. A vector field on 𝛾(𝑡), 𝑋(𝑡) is called parallel transport if the following equation is satisfied

A vector 𝑋 is a directional derivative acting on 𝑓 ∈ ℱ(𝑀) as 𝑋 ∶ 𝑓 → 𝑋(𝑓). However, there is no directional derivative

𝛻𝛾̇ (𝑡) 𝑋(𝑡) = 0, ∀ 𝑡 ∈ (𝑎, 𝑏).

acting on a tensor field of type (𝑝, 𝑞) which arises naturally from

the differentiable structure of 𝑀. What we need is an extra

Here 𝛾̇(𝑡) is the tangent vector to 𝛾(𝑡) at the point 𝑡.

structure called the connection, which how tensor are transported

along a curve.

Theorem 1.15 [4] Let 𝛾: (𝑎, 𝑏) → 𝑀 be a smooth curve on a

IJSER Definition 1.10 [3] Let 𝑀 be a smooth 𝑛-dimensional manifold,

ℱ(𝑀) be the set of smooth functions and 𝔛(𝑀) be the vector space of smooth vector fields. An affine connection on 𝑀 is a map which is denoted by 𝛻 and defined by

𝛻 ∶ 𝔛(𝑀) × 𝔛(𝑀) → 𝔛(𝑀)

manifold 𝑀. For each 𝑡0 ∈ (𝑎, 𝑏) and for each 𝑋0 ∈ 𝑇𝛾(𝑡0)𝑀 , prove that there exist a unique vector field 𝑋(𝑡) on 𝛾(𝑡) such that

(a) 𝑋(𝑡) is parallel,

(a) 𝑋(𝑡0) = 𝑋0.

Such that ( 𝑋 , 𝑌 ) ↦ 𝛻𝑋 𝑌 (a) 𝛻𝑋 (𝑌1 + 𝑌2) = 𝛻𝑋𝑌1 + 𝛻𝑋𝑌2

Proof. Let (𝑈, 𝜑) be a coordinate chart on a manifold 𝑀 at 𝑡0 with coordinates ( 𝑥1, 𝑥2, … , 𝑥𝑛). Then

(a) A smooth curve 𝛾 ∶ (𝑎, 𝑏) → 𝑀 is given by a set on 𝑛 smooth functions

(b) 𝛻𝑋1+𝑋2 𝑌 = 𝛻𝑋1𝑌 + 𝛻𝑋2𝑌 (c) 𝛻𝑋 (𝑓 𝑌) = 𝑋(𝑓) 𝑌 + 𝑓 𝛻𝑋𝑌 (d) 𝛻𝑓 𝑋𝑌 = 𝑓 𝛻𝑋𝑌, ∀ 𝑓 ∈ ℱ(𝑀) and 𝑋 , 𝑌 ∈ 𝔛(𝑀).

𝑥1 = 𝑥1(𝑡)

𝑥2

=

𝑥2(𝑡)

⎫ ⎪

:

⇒ 𝑥𝑖 = 𝑥𝑖(𝑡),

:

⎬ ⎪

𝑥𝑛 = 𝑥𝑛(𝑡) ⎭

Definition 1.11 Let (𝑈, 𝜑) be a coordinate chart on a manifold 𝑀 with a coordinates (𝑥1, 𝑥2, … , 𝑥𝑛). The functions Γikj(𝑥) are

called coordinate symbols of the affine connection 𝛻. Here Γikj(𝑥) is a 𝑛3 function, where 𝑖, 𝑗 = �1��,�𝑛�.

where 𝑡 ∈ (𝑎, 𝑏), 𝑖 = 1,2, … , 𝑛 (a) The vector 𝑋(𝑡) is given by 𝑋(𝑡) = ∑𝑛𝑖=1 𝑋𝑖 (𝑡) 𝜕𝜕𝑥𝑖 for

Definition 1.12 The vector field 𝛻𝑋 𝑓 is often called covariant derivative of vector field 𝑓 ∈ ℱ(𝑀) along the vector field 𝑋. It is

to define the covariant derivative of 𝑓 by the ordinary directional derivative,

= 𝑋1(𝑡)

𝜕 + 𝑋2(𝑡) 𝜕 +

𝜕𝑥1

𝜕𝑥2

⋯ + 𝑋𝑛(𝑡) 𝜕 .

𝜕𝑥𝑛

𝛻𝑋 𝑓 = 𝑋(𝑓).

Then 𝛾̇(𝑡) is given by

For any 𝑓, 𝑌, it can be defined as follows 𝛻𝑋 (𝑓 𝑌) = (𝛻𝑋 𝑓)𝑌 + 𝑓 𝛻𝑋 𝑌.

𝛾̇(𝑡) = ∑𝑛𝑖=1 𝑑𝑑𝑥𝑡𝑖 𝜕𝜕𝑥𝑖

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= 𝑑𝑑𝑥𝑡1 . 𝜕𝜕𝑥1 + 𝑑𝑑𝑥𝑡2 . 𝜕𝜕𝑥2 + ⋯ + 𝑑𝑑𝑥𝑡𝑛 . 𝜕𝜕𝑥𝑛 .

= 𝑓𝐴� 𝑥1, 𝑥2, … , 𝑥𝑝�

Now we have,

for any function 𝑓 ∈ ℱ(𝑀) and 𝑥1, 𝑥2, … , 𝑥𝑝 ∈ 𝔛(𝑀).

𝛻𝛾̇ (𝑡) 𝑋(𝑡) = 𝛻 𝑛 𝑑𝑥𝑖 ∑𝑛𝑗=1 𝑋𝑗 𝑒𝑗

∑𝑖=1 𝑑𝑡 𝑒𝑖

= ∑𝑛𝑖=1 𝑑𝑥𝑖

𝑑𝑡

𝛻𝑒𝑖 � ∑𝑛𝑗=1 𝑋𝑗 𝑒𝑗 �

= ∑𝑛𝑖=1 𝑑𝑥𝑖 � ∑𝑛𝑗=1 𝑒𝑖 ( 𝑋𝑗� 𝑒𝑗

𝑑𝑡

+ ∑𝑛𝑗=1 𝑋𝑗 𝛻𝑒𝑖 𝑒𝑗 ] = ∑𝑛𝑖=1 ∑𝑛𝑗=1 [ ( 𝑑𝑑𝑥𝑡𝑖 𝑒𝑖𝑋𝑗)𝑒𝑗 + 𝑑𝑑𝑥𝑡𝑖 𝑋𝑗 𝛻𝑒𝑖 𝑒𝑗 ]

Definition 1.17 A torsion 𝑇𝛻 of an affine connection 𝛻, is a map 𝑇𝛻 ∶ 𝔛(𝑀) × 𝔛(𝑀) → 𝔛(𝑀)

(𝑋, 𝑌) ↦ 𝑇𝛻 (𝑋, 𝑌) where 𝑇𝛻 (𝑋 , 𝑌) = 𝛻𝑋 𝑌 − 𝛻𝑌 𝑋 − [ 𝑋, 𝑌 ]. Theorem 1.18 For all affine connection 𝛻 , its torson 𝑇𝛻 is a tensor of rank (1, 2).

= ∑𝑛𝑖=1 ∑𝑛𝑗=1 [ ( 𝑑𝑑𝑥𝑡𝑖 𝜕𝜕𝑋𝑥𝑖𝑗) 𝜕𝜕𝑥𝑗 + 𝑑𝑑𝑥𝑡𝑖 𝑋𝑗 ∑𝑛𝑘=1 ⎾𝑖𝑘𝑗 (𝑥) 𝜕𝜕𝑥𝑘 ]

= ∑𝑛𝑗=1 𝑑𝑑𝑋𝑡𝑗 . 𝜕𝜕𝑥𝑗 + ∑𝑛𝑖,𝑗,𝑘=1 𝑑𝑑𝑥𝑡𝑖 𝑋𝑗 ⎾𝑖𝑘𝑗 (𝑥) 𝜕𝜕𝑥𝑘

Proof. We need to prove that 𝑇𝛻 (𝑓 𝑋 , 𝑌) = 𝑇𝛻 ( 𝑋 , 𝑓 𝑌 ) = 𝑓 𝑇𝛻 (𝑋 , 𝑌), ∀ 𝑓 ∈ ℱ(𝑀) and 𝑋 , 𝑌 ∈ 𝔛(𝑀).

By the definition of torsion, we get

𝑇𝛻 (𝑓 𝑋 , 𝑌) = 𝛻𝑓 𝑋 𝑌 − 𝛻𝑌 𝑓 𝑋 − [ 𝑓 𝑋, 𝑌 ]

⇒ 𝛻𝛾̇ (𝑡) 𝑋(𝑡) = ∑𝑛𝑘=1 [ 𝑑𝑋𝑘

𝑑𝑡

= 𝑓 𝛻𝑋 𝑌 − 𝑌(𝑓)𝑋 − 𝑓 𝛻𝑌 𝑋 − [ 𝑓 𝑋, 𝑌 ]

IJSER + ∑𝑛𝑖,𝑗=1 𝑑𝑑𝑥𝑡𝑖 𝑋𝑗 ⎾𝑖𝑘𝑗(x)] 𝜕𝜕𝑥𝑘.

Thus the equation for the parallel transport is 𝑑𝑑𝑋𝑡𝑘 + ∑𝑛𝑖,𝑗=1 𝑑𝑑𝑥𝑡𝑖 𝑋𝑗 Γikj(𝑥) = 0 �

𝑋𝑘(𝑡0) = 𝑋𝑘0 (𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛)

… … … Now take any g ∈ ℱ(𝑀), then [ 𝑓 𝑋, 𝑌 ]g = 𝑓 𝑋� 𝑌(g)� − 𝑌(𝑓 𝑋(g))

= 𝑓 𝑋𝑌(g) − 𝑌(𝑓). 𝑋(g) − 𝑓 𝑌𝑋(g) = 𝑓 � 𝑋𝑌(g) − 𝑌𝑋(g)� − 𝑌(𝑓). 𝑋(g)

This is a system of 𝑛-equations for 𝑛-unknown functions 𝑋𝑘(𝑡) with 𝑛-initial conditions. A theorem from the theory of

= 𝑓 [𝑋, 𝑌]g − 𝑌(𝑓). 𝑋(g)

differential equations says that the solution exists and which is unique. This completes the proof of this theorem.

∴ [ 𝑓 𝑋, 𝑌 ] = 𝑓 [𝑋, 𝑌] − 𝑌(𝑓) 𝑋.

(1.01)

Therefore, equation (1.01) becomes

V. Torsion tensor and Riemann curvature tensor

𝑇𝛻 (𝑓 𝑋 , 𝑌) = 𝑓 𝛻𝑋 𝑌 − 𝑌(𝑓)𝑋 − 𝑓 𝛻𝑌 𝑋 − 𝑓 [𝑋, 𝑌]

In the mathematical field of differential geometry, the Riemann curvature tensor, or Riemannian–Christoffel tensor is the most standard way to express curvature of Riemannian manifolds. It associates a tensor to each point of a Riemannian manifold that measures the extent to which the metric tensor is not locally isometric to a Euclidean space.

Again,

+𝑌(𝑓) 𝑋 = 𝑓 ( 𝛻𝑋 𝑌 − 𝛻𝑌 𝑋 − [𝑋, 𝑌] ) = 𝑓 𝑇𝛻 (𝑋 , 𝑌).

Definition 1.16 [5] Let 𝑀 be a smooth 𝑛-dimensional manifold, ℱ(𝑀) be the set of smooth functions and 𝔛(𝑀) be the vector space of smooth vector fields. A tensor 𝐴 of rank (1, 𝑝) on 𝑀 is a multi-linear map

𝐴 ∶ 𝔛(𝑀) × 𝔛(𝑀) × … × 𝔛(𝑀) → ℱ(𝑀)

which satisfies

𝐴�𝑓 𝑥1, 𝑥2, … , 𝑥𝑝� = 𝐴� 𝑥1, 𝑓𝑥2, … , 𝑥𝑝� = ⋯

𝑇𝛻 ( 𝑋 , 𝑓𝑌) = − 𝑇𝛻 ( 𝑓𝑌 , 𝑋)

[ ∵ 𝑇𝛻 (𝑋 , 𝑌) = − 𝑇𝛻 (𝑌 , 𝑋)] = − 𝑓 𝑇𝛻 ( 𝑌 , 𝑋) [as previous part] = 𝑓 𝑇𝛻 (𝑋 , 𝑌).

Therefore, 𝑇𝛻 is a tensor of rank (1, 2). Hence completes the proof.

= 𝐴� 𝑥1, 𝑥2, … , 𝑓𝑥𝑝�

Definition 1.19 The curvature tensor 𝑅𝛻, of an affine connection 𝛻 , is a map

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𝑅𝛻 ∶ 𝔛(𝑀) × 𝔛(𝑀) × 𝔛(𝑀) → 𝔛(𝑀)

+𝑓 𝛻𝑋 𝛻𝑌 𝑍 − 𝑌�𝑋(𝑓)�𝑍 − 𝑋(𝑓)𝛻𝑌 𝑍

(𝑋, 𝑌, 𝑍) ↦ 𝑅𝛻 (𝑋, 𝑌, 𝑍)

− 𝑌(𝑓)𝛻𝑋 𝑍 − 𝑓 𝛻𝑌 𝛻𝑋 𝑍 − [𝑋, 𝑌]𝑓. 𝑍

where 𝑅𝛻 (𝑋, 𝑌, 𝑍) = 𝛻𝑋 𝛻𝑌 𝑍 − 𝛻𝑌 𝛻𝑋 𝑍 −𝛻[ 𝑋,𝑌 ]𝑍.

−𝑓 𝛻[ 𝑋,𝑌 ] 𝑍

Theorem 1.20 [6] For all affine connection 𝛻, its curvature 𝑅𝛻 is a tensor of rank (1, 3).

Proof: We need to prove that

𝑅𝛻 (𝑓 𝑋 , 𝑌, 𝑍) = 𝑅𝛻 ( 𝑋 , 𝑓 𝑌, 𝑍 ) = 𝑅𝛻 ( 𝑋 , 𝑌, 𝑓𝑍 ) = 𝑓 𝑅𝛻 (𝑋 , 𝑌, 𝑍); ∀ 𝑓 ∈ ℱ(𝑀) and 𝑋 , 𝑌, 𝑍 ∈

𝔛(𝑀). By the definition of curvature tensor, we get

𝑅𝛻 (𝑓 𝑋, 𝑌, 𝑍)

= 𝛻𝑓 𝑋 𝛻𝑌 𝑍 − 𝛻𝑌 𝛻𝑓𝑋 − 𝛻[ 𝑓 𝑋,𝑌 ] 𝑍

= 𝑓 𝛻𝑋 𝛻𝑌 𝑍 − 𝛻𝑌 ( 𝑓 𝛻𝑋 𝑍 ) − 𝛻𝑓[ 𝑋,𝑌 ]–𝑌(𝑓)𝑋 𝑍

= �𝑋𝑌(𝑓) − 𝑌𝑋(𝑓)�𝑍 + 𝑓(𝛻𝑋 𝛻𝑌 𝑍−𝛻𝑌 𝛻𝑋 𝑍 −𝛻[ 𝑋,𝑌 ] 𝑍) − [𝑋, 𝑌]𝑓. 𝑍

= [𝑋, 𝑌]𝑓. 𝑍 + 𝑓 𝑅𝛻 (𝑋 , 𝑌, 𝑍) − [𝑋, 𝑌]𝑓. 𝑍

= 𝑓 𝑅𝛻 (𝑋 , 𝑌, 𝑍).

Therefore, 𝑅𝛻 is a tensor of rank (1, 3). Hence completes the proof.

VI. Levi-Civita connections

Let 𝑀 be a smooth n-dimensional manifold, ℱ(𝑀) be the set of smooth functions, g be a smooth metric, 𝔛(𝑀) be the vector space of smooth vector fields and 𝛻 be an affine connection on 𝑀. Then the covariant derivative on g with respect to 𝛻 is a multilinear map,

= 𝑓 𝛻𝑋 𝛻𝑌 𝑍 − 𝑌(𝑓) 𝛻𝑋 𝑍 − 𝑓 𝛻𝑌 𝛻𝑋 𝑍 − 𝛻𝑓[ 𝑋,𝑌 ] 𝑍

𝛻g ∶ 𝔛(𝑀) × 𝔛(𝑀) × 𝔛(𝑀) → ℱ(𝑀)

IJSER + 𝛻𝑌(𝑓)𝑋𝑍

= 𝑓 𝛻𝑋 𝛻𝑌 𝑍 − 𝑌(𝑓) 𝛻𝑋 𝑍 − 𝑓 𝛻𝑌 𝛻𝑋 𝑍 − 𝑓 𝛻[ 𝑋,𝑌 ] 𝑍 +𝑌(𝑓) 𝛻𝑋 𝑍

= 𝑓 ( 𝛻𝑋 𝛻𝑌 𝑍 − 𝛻𝑌 𝛻𝑋 𝑍 − 𝛻[ 𝑋,𝑌 ] 𝑍 )

= 𝑓 𝑅𝛻 (𝑋 , 𝑌, 𝑍).

Again, note that

( 𝑍, 𝑋 , 𝑌 ) ↦ 𝛻𝑍g(𝑋, 𝑌) where 𝛻𝑍g(𝑋, 𝑌) = 𝑍g(𝑋, 𝑌) − g(𝛻𝑍𝑋, 𝑌) − g(𝑋, 𝛻𝑍𝑌)

Definition 1.21 [7] Let M be a smooth manifold equipped with a smooth manifold metric g. There is a unique affine connection 𝛻 on g such that

(a) 𝛻 is torsion free.

𝑅𝛻 ( 𝑋, 𝑌, 𝑍) = − 𝑅𝛻 ( 𝑌, 𝑋, 𝑍) Then

𝑅𝛻 (𝑋, 𝑓 𝑌, 𝑍) = − 𝑅𝛻 (𝑓 𝑌, 𝑋, 𝑍) = − 𝑓 𝑅 𝛻 (𝑌, 𝑋, 𝑍) [as previous part]

(b) 𝛻g = 0 This unique connection is called Levi-Civita connection.

Theorem 1.22 (The fundamental theorem of pseudo-Riemannian geometry) On a pseudo-Riemannian manifold (𝑀, g), there exists a unique symmetric connection (Levi-Civita connection) which is compatible with the metric g.

Also,

= 𝑓 𝑅𝛻 (𝑋 , 𝑌, 𝑍).

Proof: Let 𝛾 be a tangent vector to an arbitrary curve along which the vectors are parallel transported. Then we have,

𝑅𝛻 (𝑋, 𝑌, 𝑓 𝑍) = 𝛻𝑋 𝛻𝑌 𝑓 𝑍 − 𝛻𝑌 𝛻𝑋 𝑓 𝑍 − 𝛻[ 𝑋,𝑌 ] 𝑓 𝑍

= 𝛻𝑋 (𝑌(𝑓)𝑍 + 𝑓 𝛻𝑌 𝑍 − 𝛻𝑌 ( 𝑋(𝑓). 𝑍 + 𝑓 𝛻𝑋 𝑍) −[𝑋, 𝑌](𝑓). 𝑍 𝑓 𝛻[ 𝑋,𝑌 ] 𝑍

= 𝛻𝑋 ( 𝑌(𝑓)𝑍 ) + 𝛻𝑋 (𝑓 𝛻𝑌 𝑍 ) − 𝛻𝑌 (𝑋(𝑓). 𝑍) −𝛻𝑌 ( 𝑋(𝑓). 𝑍)− 𝛻𝑌 (𝑓 𝛻𝑋 𝑍) − [𝑋, 𝑌](𝑓). 𝑍 −𝑓 𝛻[ 𝑋,𝑌 ] 𝑍

= 𝛻𝑋 ( 𝑌(𝑓)𝑍 ) + 𝑌(𝑓) 𝛻𝑋 𝑍 + 𝑋(𝑓)𝛻𝑌 𝑍

0 = ∇𝛾[g(X, Y)] = γi[(∇i g)(X, Y)] + g(∇iX, Y) +g(X, ∇iY)

= γi 𝑋𝑗𝑌𝑘(∇𝑖g)𝑗𝑘).

where we have noted that ∇i X = ∇i Y = 0. Since this true for any curves and vectors, we must have

(∇ig)jk = 0.

For metric tensor we know,

(∇ℓg)jk = ∂∂xℓ gjk − Γℓi jgik − Γℓi kgij. Now from the above, we can write as follows

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∂∂xℓ gjk − Γℓi jgik − Γℓi kgij = 0 … … … (1.02)

Levi-Civita

connection

coefficients

are

Γ

𝜙 𝑟𝜙

=

Γ

𝜙 𝑟𝜆

=

𝑟−1

and

Γ

𝑟 𝜙𝜙

= −𝑟.

Now using cyclic permutations of (ℓ, 𝑗, 𝑘), we have ∂∂xj gkℓ − Γij kgiℓ − Γij ℓ gik = 0 … … … (1.03)

∂∂xk gℓj − Γikℓ gij − Γikjgiℓ = 0 … … … (1.04)

Example 1.24 The induce map on 𝑆2 is 𝑔 = 𝑑𝜃⨂𝑑𝜃 + 𝑠𝑖𝑛2𝜃 𝑑𝜙 ⨂ 𝑑𝜙. the non-vanishing components of the LeviCivita connections are

Γ

𝜃 𝜙𝜙

= −𝑐𝑜𝑠𝜃 𝑠𝑖𝑛𝜃 ;

Γ

𝜙 𝜃𝜙

=

Γ

𝜙 𝜙𝜃

= 𝑐𝑜𝑡𝜃.

The combination −(1.02) + (1.03) + (1.04) yields

− ∂∂xℓ gjk + ∂∂xj gkℓ + ∂∂xk gℓj + Τℓi jgik +Τiℓ kgij − 2Γi(j k)giℓ = 0 … … … (1.05)

where Τℓi j = 2Γ[iℓ j] = Γℓi j − Γij ℓ and Γi(j k) = 12 �Γki j + Γji k �. The

tensor

Τ

i ℓ

j

is

anti-symmetric

with

respect

to

the

lower

indices

Τℓi j = −Τij ℓ.

By solving equation (1.05) for Γi(j k), we have

VII. Killing Vector Fields

A Killing vector field is a vector field on a Riemannian manifold that preserves the metric. Killing fields are the infinitesimal generators of isometrics, that is, flows generated by Killing fields are continuous isometrics of the manifold.

Definition 1.25 [9] A vector field 𝑋 is a Killing vector field if the Lie derivative with respect to 𝑋 of the metric 𝑔 vanishe

ℒ𝑥𝑔 = 0.

In terms of the Levi-Civita connection, this is

Γ

i (j

k

)

=

�

i

�

+

1

�Τ

i k

j

+

Τ

i j

k

�

…

…

…

(1.06)

jk 2

𝑔( 𝛻𝑌 𝑋, 𝑍) + 𝑔(𝑌, 𝛻𝑍 𝑋) = 0

IJSER where �jik� are the christoffel symbols defined by

�𝑗𝑖𝑘� = 12 𝑔𝑖 𝑙 �𝜕𝜕𝑔x𝑘j𝑙 + 𝜕𝜕𝑔x𝑗k𝑙 − 𝜕𝜕𝑔x𝑗ℓ𝑘� … … … (1.07)

Finally, the connection coefficient Γ is given by

Γ

i j

k

=

Γ

i (j

k)

+

Γ

i [j

k]

for all vectors 𝑌 and 𝑍. In local coordinates, this amounts to the Killing equation

𝛻𝑖𝑋𝑗 + 𝛻𝑗𝑋𝑖 = 0.

This condition is expressed in covariant form. Therefore it is sufficient to establish it in a preferred coordinate system in order to have it hold in all coordinate systems.

Examples 1.26 The vector field on a circle that points clockwise

=

�𝑗𝑖𝑘�

+

1 2

�Τki

j + Τji

k + Τij k�

…

…

… (1.08)

and has the same length at each point is a Killing vector field, since moving each point on the circle along this vector field

The second term of the last expression of (1.08) is called contorsion, denoted by Ki jk:

simply rotates the circle.

VIII. Conformal Killing Vector Fields

𝐾𝑖 = 1 (Τi + Τi + Τi )

𝑗𝑘 2 k j j k

j k

The term of the conformal Killing vector field is an extension of the term of the Killing vector field. Conformal Killing vector

So,

Γ𝑗𝑖𝑘

=�

𝑖

�

+

𝐾

𝑖 𝑗

𝑘

𝑗 𝑘

Now,

Γ

𝑖 (𝑗

𝑘)

=

Γ

𝑖 𝑗

𝑘

+

𝑇

𝑖 𝑗

𝑘

is the another connection coefficient

if 𝑇 is a tensor field of type (1,2). Now we choose

𝑇

𝑖 𝑗

𝑘

=

−𝐾

𝑖 𝑗

𝑘

so that

Γ

𝑖 𝑗

𝑘

=�

𝑖

�

𝑗 𝑘

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 transformations.

Definition 1.27 [10] Let (𝑀, 𝑔) be a Riemannian manifold and 𝑋 ∈ 𝔛(𝑀). Then the vector field 𝑋 is a conformal Killing vector field, if an infinitesimal displacement given by 𝜀 𝑋 generates a conformal transformation.

Example 1.28 Let 𝑥𝑘 be the coordinates of (ℝ𝑚, 𝛿). The vector

= 12 𝑔𝑖 𝑙 �𝜕𝜕𝑔𝑥𝑙𝑗𝑘 + 𝜕𝜕𝑔𝑥𝑗𝑘𝑙 − 𝜕𝜕𝑔𝑥𝑗𝑙𝑘�. By construction, this is symmetric and certainly unique given a

𝐷 = 𝑥𝑘 𝜕𝜕𝑥𝑘 is a conformal killing vector.

metric.

IX. Conclusion

Example 1.23 Let metric on ℝ2 in polar coordinates is 𝑔 = 𝑑𝑟 ⨂ 𝑑𝑟 + 𝑟2𝑑𝜙 ⨂ 𝑑𝜙. The non-vanishing components of the

The fundamental theorem of pseudo-Riemannian geometry is established using tensors on a manifold 𝑀. In this theorem, I have

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used metric connection ∇ which is the natural generalization of

the connection defined in the classical geometry of surfaces. The

covariantly constant metric 𝑔𝑖𝑗 and vectors fields 𝑋 and 𝑌, which

are parallel transported along any curve are used in this theorem.

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.

References

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[2] Auslander, L. and Mackenzie, R. E. 1963. Introduction to differential manifolds, McGra Hill, New York.

[3] Ahmed, K. M. 2003. Curvature of Riemannian Manifolds, The Dhaka University Journal of Science, 25-29.

[4] Islam, J. Chrish, 1989. Modern Differential Geometry for Physicists, World Scientific Publishing Co. Pte. Ltd.

[5] Ahmed, K. M. 2003. A note on affine connection on Differetial Manifolds, The Dhaka University Journal of

IJSER Science, 63-68.

[6] Choquer-Bruhat, Y. and De Witt-Morette, C. DillardBleick, M. 1982. Analysis Manifolds and Physics edition, Amterdam, North-Holland.

[7] Brickell, F. and Clark, R. S. 1970. Differential Manifolds, Van Nostrand Reinhold Company, London

[8] Bott, R. and Tu, L. W. 1982. Different forms in Algebric Topology, New York.

[9] Chevally, C. 1956 Fundamental concepts of Algebra, Academic Press, New York.

[10] Donson, C. T. J. and Poston, T. 1977. Tensor Geometry, Pitman, London,.

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