Pseudo - Riemannian

Geometry

and

Tensor Analysisby

Rolf Sulanke

Started February 1, 2015.

Version 1 finished May 20, 2016.

Version 2 started March 30, 2017

Version 2 finished July 10, 2017

Mathematica v. 9.0.1,, 10.4.1, 11.1.1

Preface

In this notebook I develop and explain Mathematica tools for applications to Riemannian

geometry and relativity theory. Together with the notebooks concerned with Euclidean

geometry of curves and surfaces in [EDG] it may be used as an interactive introductory

textbook of differential geometry. Clearly, these notebooks don’t cover the full content of an

introductory course for this field. Therein the applications of algebra and calculus to differen-

tial geometry, the algorithmic methods are emphasized, and the rich graphical tools of

Mathematica are applied to illustrate the concepts in dimensions 2 and 3. In order to learn

differential geometry, the notebooks will give best profits when using them as an accompany-

ing tool to an introductory course or to studying one of the numerous textbooks. Despite of

this seemingly restrictive recommendation I tried to compose a coherent textbook contain-

ing the fundamental concepts of differential geometry understandable for users with a basic

knowledge of calculus, linear and tensor algebra. The tools of tensor algebra as applied in

this notebook can be found and downloaded here, they are included in the packed file

RG.zip. Furthermore I hope that the fine structured presentation of the content, the tables

of keywords and usages, and the open access to the program sources written as Mathemat-

ica packages will be useful for researchers and engineers who need to apply differential

geometric methods in their work. This open access may also be useful for those striving to

learn programming their own Mathematica functions or to test and, if possible, to improve

the proposed programs.

The contents of the notebook can be described shortly as follows. Section 1, based on

concepts and functions contained in the notebook euklsfv3.nb of Euclidean surface geome-

try, treats the Riemannian spaces of constant curvature. They serve as standard examples

in the following sections.

Section 2 starts with the definition of differentiable manifolds and its tangent, cotangent, and

tensor bundles. I introduce the calculus of differential forms and the corresponding Mathe-

matica functions and operations which form the exterior algebra including the exterior

differential operator, a calculus widely used in geometry, physics and topology. This exterior

calculus exists on each differentiable manifold independently of any additional structure.

Differential forms are antisymmetric covariant tensor fields, and only for such fields the

exterior differential is defined. The only differential operator for vector fields existing in such

generality is the commutator of two fields, treated in Subsection 2.4.9.

Section 2 starts with the definition of differentiable manifolds and its tangent, cotangent, and

tensor bundles. I introduce the calculus of differential forms and the corresponding Mathe-

matica functions and operations which form the exterior algebra including the exterior

differential operator, a calculus widely used in geometry, physics and topology. This exterior

calculus exists on each differentiable manifold independently of any additional structure.

Differential forms are antisymmetric covariant tensor fields, and only for such fields the

exterior differential is defined. The only differential operator for vector fields existing in such

generality is the commutator of two fields, treated in Subsection 2.4.9.

In Section 3 is shown that the partial derivatives of the coordinates of a vector field can not

be considered as the coordinates of a tensor. To get a gradient tensor or a differential of

arbitrary tensor fields an additional structure must be given on the manifold, an affine or,

more precisely, a linear connection. The theory of linear connections is developed in this

Section, gradients and absolute differentials of tensor fields, torsion and curvature tensors

of the linear connection are defined. The geometric background of this structure is the

question whether there exist absolutely parallel tensor fields, whose gradient is a null

tensor. Geodesics are defined as autoparallel curves.

Section 4 is devoted to pseudo-Riemannian manifolds. The theorem of Levi-Civita is

proved: On each such manifold there exists a uniquely defined linear connection with

torsion tensor zero for which the absolute differential of the metric tensor is zero. The

special properties of this Levi-Civita connection are deduced. Lowering and raising of

indices yield canonical isomorphisms between tensor spaces of distinct covariance types of

the same degree. Using the duality of vector and tensor spaces canonically defined scalar

products in these spaces are introduced. The special properties of the Riemannian curva-

ture tensor, the symmetries of its covariant form, the Bianchi identity are considered. The

sectional curvature is introduced and the Gaussian curvature of surfaces is identified as its

special case.

The second version of the notebook is - in difference to the first version - now separated

from the elementary Euclidean differential geometry; it can be read and applied indepen-

dently of these elementary part. All necessary packages and notebooks are collected in the

packed file RG.zip. I included also the notebook vectensalgv3.nb containing an implementa-

tion of tensor algebra into Mathematica as it is used in the current notebook. Since many

definitions in the new versions are changed or are new, the Mathematica packages of

different versions should not be mixed; use exactly those contained in RG.zip.

I conclude this preface expressing my sincere thanks to the Humboldt University in Berlin

for the continuous support giving me access to Wolfram's Mathematica and the possibility to

continue my work also in my retired position.

Keywords

absolute differential index raising

absolute parallel inner geometry

affine connection inverse metric

algebra Lie inverse metric tensor

algebra of functions isotropic

alternated absolute differential Jacobi

antisymmetric tensors Jacobian

arclength Kronecker symbol

atlas Levi-Civita connection

auto-parallel curves linear connection

Bianchi’s identity local coordinates

bivector Lorentzian space

category manifolds

chart metric

Christoffel symbols metric tensor

class Ck multi indices

commutator nabla

complete atlas n-sphere

commutator orientable manifold

2 RGv2.nb

conformal orientation

connection forms oriented atlas

contraction parallel displacement

contravariant p-form

coordinate transformation product rule

covariant curvature tensor pseudo-Euclidean space

covector fields pseudo-Riemannian space

curvature tensor regular

curve Ricci tensor

diffeomorphism Riemannian space

differential Riemannian spaces of constant curvature

differential form sectional curvature

dimension singular

directional derivative smooth

duality spacelike

equi sphere

equivalent tensors sphere of imaginary radius

Euclidean metric stereographic projection

exterior algebra strict coordinates

exterior differential submanifold

exterior product sum convention

fiber surface

fiber type symmetric tensors

flat linear connection tangent basis

G-symbols tangent bundle

Gaussian curvature tangent space

Gradient tangent vector

gradient tensor tensor fields

Hausdorff space tensor metric

homeomorphism tensor product

hyperbolic space timelike

hyperboloid torsion tensor

hypersurface traditional notation

immersiom vector fields

index wedge

index lowering wedge product

keylist

Copyright

This notebook and the accompanying packages are public. Authors who intend to publish a

changed or completed version of them should do it under their own names with the condi-

tion that they cite the original notebook with the Internet address or other source where they

got it. I am not responsible for errors or damages originated by the use of the procedures

contained in my notebooks or packages; everybody who applies them should test carefully

whether they are appropriate for his purposes.

Initialization

Imported Functions and their Usages

RGv2.nb 3

Imported Functions and their Usages

This Subsection contains tables of the symbols defined in the imported packages and in the

Global context.

Hints

Symbol Tables. Usages.

Riemannian Geometry and Linear Connections

Gaussian Curvature

Euclidean Linear Algebra

Tensor Algebra

Pseudo-Euclidean Linear Algebra

Alfred Gray’s Catalogue of Curves and Surfaces

The Global Context

1. Riemannian Spaces of Constant Curvature

In this Section we introduce n-dimensional Riemannian metrics of constant curvature. The

manifolds considered here are open, connected sets of an n-dimensional Euclidean space

provided with orthonormal Cartesian coordinates. More general manifold concept will be

treated in Subsection 2.1.

1.1. Pseudo-Riemannian Metrics

1.1.1. A General pseudo-Riemannian Metric

1.1.2. Inner Geometry of an Immersion

1.1.3. The Inverse Metric Matrix

1.1.4. The Euclidean Metric

1.2. Riemannian Spaces of Constant Positive Curvature

1.3. Riemannian Spaces of Constant Negative Curvature

1.3.1. Spheres of imaginary Radius (n-Pseudo-Spheres).

1.3.2. Conformal Model of the hyperbolic n-spaces

2. Tensor Fields

In this and the following Sections we often use the functions and modules of tensor algebra

contained in the package tensalgv3.m. To learn this matter I recommend first to study the

notebook vectensalgv3.nb.

4 RGv2.nb

In this and the following Sections we often use the functions and modules of tensor algebra

contained in the package tensalgv3.m. To learn this matter I recommend first to study the

notebook vectensalgv3.nb.

2.1. Manifolds

2.1.1. The Definition

2.1.2. The Rn and the n-Spheres S

n

2.1.3. The Category of Differentiable Manifolds

2.2.2 Tangent Bundle

2.2.3. Cotangent Bundle

2.3. Vector and Tensor Fields on a Manifold

2.3.1. Definition of Tensor Fields

2.3.2. Functions as Tensor Fields of Type {0, 0}

2.3.3. Vector Fields

2.3.4. Directional Derivatives

2.3.5. Local Bases. Commuting Properties

2.4. Exterior Differential in Tensor Description

Mathematica functions modelling the exterior differential calculus for p-forms and antisym-

metric covariant tensors are contained in the package eudiffgeov3.m. Their applications to p-

forms described in the notebook euklsfv3.nb in [EDG]. The exterior differential is denoted by

dd.

It remains to consider the exterior differential in the realm of tensors.

? dd

2.4.1. The Exterior Algebra of Antisymmetric Covariant Tensor

Fields

2.4.2. The Exterior Differential of Antisymmetric Arrays in n

Dimensions

2.4.3. The Exterior Differential of Antisymmetric Covariant Tensors

2.4.4. Linearity and Product Rules

2.4.5. Exterior Differential of 1-Forms and Commutator of Vector

Fields

3. Absolute Differentials and Gradients of Vector

and Tensor Fields

3.1. The Differentiation of Vector Fields

RGv2.nb 5

3.2. Linear Connections

3.2.1. Introduction

3.2.2. The Gradient of a Vector Field

3.2.3. The Absolute Differential of Functions, Vector, and Covector

Fields

3.2.4. The Absolute Differential of a Tensor Field of Type (p, q)

3.2.5. The Absolute Differential, Modules absD, tensorD

3.2.6. Properties of the Absolute Differential

3.3. The Gradient Tensor

3.3.1. Definition and Examples

3.3.2. Properties of Gradient Tensors

3.4. Parallelism

3.4.1. Absolute Parallel Vector Fields

3.4.2. Parallel Displacement Along Curves

3.4.3. Auto-Parallel Curves

3.5. Properties of the Curvature and Torsion Tensors

3.5.1. Symmetry Properties

3.5.2. Alternated Twofold Absolute Derivations

3.5.2.1. Functions

3.5.2.2. Curvature and Torsion Operators

3.5.2.3. T(X,Y)

3.5.2.4. R(X,Y). Vector Fields

3.5.2.5. A Covector Field on the Sphere

3.5.2.6. Covector Fields

3.5.3. The Ricci Tensor

4. Pseudo-Riemannian Spaces

4.1. The Levi-Civita Connection

6 RGv2.nb

4.2. The Curvature Tensor

4.2.1. Ricci’s Lemma

4.2.2. Curvature of Immersions. Examples

4.2.3. The Covariant Curvature Tensor

4.2.4. Duality and Tensor Metric

4.2.5. Sectional Curvature

Epilog

In this Epilog I mention some important subjects not treated in the notebooks; the user may

continue the study of differential geometry in one or several of the mentioned directions and

elaborate Mathematica functions for the corresponding calculations. The references given in

the epilog as well as in the Notebooks are very subjective and in no way complete. Please

use the mathematical databases Mathematical Reviews or Zentralblatt to find the rich actual

literature. Furthermore I recommend to look into the Wolfram Library Archive to find informa-

tion on Mathematica projects and packages for the interested item. In particular, search

there for “Tensor” tom find information and sources on work based on Mathematica, e. g.

MathTensor, H. Soleng, José M. Martín-García, and others.

1. Relativity Theory

The space-time world of special relativity theory is the Lorentz space: the four-dimensional

pseudo-Euclidean vector space of index 1, and that of the general relativity theory is a 4-

dimensional pseudo-Riemannian manifold of index 1. Thus the Mathematica concepts

contained in this notebook and the accompanying packages may serve as a starting point to

go ahead into this field. There exists a lot of applications of Mathematica to this field. Many

concrete calculations cam be found in the work of Sortirios Bonanos who unfortunately

already died in 2013. Further see the commercial package Cartan of Harald. H. Soleng, the

package ccgrg of Andrzej Woszczyna and others.

2. Lie Groups and Lie Algebras

The theory of Lie groups and algebras is needed for a wide field of developments in differen-

tial geometry, geometric analysis, and algebra with important applications to physics. There

are some special programs written in Mathematica, e. g. the work LieArt of Robert Feger

and Thomas w. Kephart, or my old package, but a systematic treatment of this theory in

Mathematica concepts seems to be missing yet. An interactive textbook developed along

the lines of S. Helgason’s book would be an important step in using Mathematica methods

in transformation groups, fiber bundles and holonomy theory, homogeneous, in particular

symmetric spaces.

3. Submanifolds of Homogeneous Spaces with Cartan Methods

Here I have in mind less the general theory but the investigation of special problems in

projective, affine, symplectic or Möbius spaces. For these fields there exists an overwhelm-

ing large number of publications, but not so much is done, outside of Euclidean geometry,

with Mathematica. Just appeared the book of G. R. Jensen, E. Musso, and L. Nicolodi

which could stimulate interesting Mathematica projects in this field. Clearly there are many

further subjects which could be mentioned, like complex geometry, geometric theory of

differential equations, Hamiltonian equations and calculus of variations, but to discuss them

here would break the limits of this notebook.

I will be very grateful for all comments, critical remarks or hints to errors. Reports on

experiences of using the notebooks in teaching differential geometry are welcome. Please,

write me an e-mail to sulanke@mathematik.hu-berlin.de.

RGv2.nb 7

I will be very grateful for all comments, critical remarks or hints to errors. Reports on

experiences of using the notebooks in teaching differential geometry are welcome. Please,

write me an e-mail to sulanke@mathematik.hu-berlin.de.

Berlin. July 10, 2017.

Rolf Sulanke

References

The References contain only citations used in this notebook. In many textbooks or mono-

graphs about differential geometry one finds comprehensive bibliographies.

The newest versions of the notebooks and packages on differential geometry are available

on my homepage:

IDG

R. Sulanke. Differential Geometry. Since 2016.

B-R

W. Blaschke, H. Reichardt. Einführung in die Differentialgeometrie.2. Auflage. Springer-

Verlag. Berlin.Göttingen.Heidelberg. 1960.

B-L

W. Blaschke, K. Leichtweiß. Elementare Differentialgeometrie. 5. Auflage. Springer-Verlag.

Berlin.Göttingen.Heidelberg. 1973.

Christ

Steven. M. Christensen & Associates. MathTensor.

Remark. This softtware seems to be not available now,

see http://community.wolfram.com/groups/-/m/t/339315

Gray

[G06] Alfred Gray, Simon Salamon, Elsa Abbena. Modern Differential Geometry of Curves

and Surfaces with Mathematica.

Third ed. CRC Press. 2006.

Gray2

[G06nb] Alfred Gray, Simon Salamon, Elsa Abbena. A series of 27 Mathematica notebooks

corresponding to the chapters of [G06], available from the website

http://webmath2.unito.it/paginepersonali/abbena/gray/

Gray3

[G94] Alfred Gray. Differentialgeometrie. Klassische Theorie in moderner Darstellung.

(Übersetzung aus dem Amerikanischen H. Gollek).

Spektrum Akademischer Verlag, Heidelberg.Berlin.Oxford. 1994.

H78

S. Helgason.Differential Geometry, Lie Groups, and Symmetric Spaces.

Academic Press, 1978.

JMN

G. R. Jensen, E. Musso, L. Nicolodi. Surfaces in Classical Geometries. A Treatment by

Moving Frames. Universitext. Springer-Verlag. 2016.

K-N1

S. Kobayashi, K. Nomizu. Foundations of Differential Geometry, Vol. I. Interscience Publish-

ers. New York - London. 1963.

Kr

Erwin Kreyszig. Differentialgeometrie. Akademische Verlagsgesellschaft Geest & Portig K.

G. Leipzig 1957.

Mal

A. I. Maltzev. Grundlagen der linearen Algebra (Russ.) , Moskau 1956.

xAct

José M.Martín - García. xAct: Efficient tensor computer algebra for the Wolfram Language.

A.L.Onishchik, Sulanke, R. Algebra und Geometrie, Einführung. VEB Deutscher Verlag der

Wissenschaften,Berlin 1977. (2. Aufl.Berlin 1986, Russ. Moscow 2004)

8 RGv2.nb

OSI

A.L.Onishchik, Sulanke, R. Algebra und Geometrie, Einführung. VEB Deutscher Verlag der

Wissenschaften,Berlin 1977. (2. Aufl.Berlin 1986, Russ. Moscow 2004)

OSII

A.L.Onishchik, Sulanke,R. Algebra und Geometrie, 2. Moduln und Algebren. VEB

Deutscher Verlag der Wissenschaften, Berlin 1988, (Russ. Moscow 2008).

OSIII

[OSIII] A.L. Onishchik, Sulanke, R. Projective and Cayley-Klein Geometries, Springer

Monographs in Mathematics, Berlin Heidelberg 2006.

Sch54

J. A. Schouten. Ricci-Calculus. Grundlehren der mathematischen Wissenschaften Vol. X,

2nd. edition. Springer-Verlag 1954.

Soleng

Harald H. Soleng. Tensors in Physics: CARTAN 1.8. December 28, 2011

Copyright © 2002-2011 by Ad Infinitum

http://arxiv.org/abs/gr-qc/9502035

EDG

[EDG] R.Sulanke. An Interactive Textbook on Euclidean Differential Geometry. Version 1.

Wolfram Library Archive 2015.

New versions: Introduction to Differential Geometry. Section Downloads, EDG.zip.

Tensors

Tutorial Symbolic Tensors

ccgrg

Andrzej Woszczyna and others. ccgrg - the symbolic tensor analysis package, with tools for

general relativity. Wolfram Library Archive. MathSource/8848.

Homepage

Home

http://www-irm.mathematik.hu-berlin.de/~sulanke

e-mail: sulanke@mathematik.hu-berlin.de

RGv2.nb 9