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Taha Sochi - Introduction to Differential Geometry of Space Curves and Surfaces

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Book:
Introduction to Differential Geometry of Space Curves and Surfaces
Author:
Taha Sochi
Publisher:
CreateSpace Independent Publishing Platform
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Books / Children
Year:
2017
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This book is about differential geometry of space curves and surfaces. The formulation and presentation are largely based on a tensor calculus approach. It can be used as part of a course on tensor calculus as well as a textbook or a reference for an intermediate-level course on differential geometry of curves and surfaces. The book is furnished with extensive sets of exercises and many cross references, which are hyperlinked, to facilitate linking related concepts and sections. The book also contains a considerable number of 2D and 3D graphic illustrations to help the readers and users to visualize the ideas and understand the abstract concepts. We also provided an introductory chapter where the main concepts and techniques needed to understand the offered materials of differential geometry are outlined to make the book fairly self-contained and reduce the need for external references.

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Preface

The present book is about differential geometry of space curves andsurfaces. The formulation and presentation are largely based on atensor calculus approach, which is the dominant trend in the modernmathematical literature of this subject, rather than the geometricapproach which is usually found in some old style books. The book isprepared, to some extent, as part of tutorials about topics andapplications related to tensor calculus. It can therefore be used aspart of a course on tensor calculus as well as a textbook or areference for an intermediate-level course on differential geometry ofcurves and surfaces.

Apart from general background knowledge in anumber of mathematical branches such as calculus, geometry and algebra,an important requirement for the reader and user of this book isfamiliarity with the terminology, notation and concepts of tensorcalculus at reasonable level since many of the notations and conceptsof differential geometry in its modern style are based on tensorcalculus.

The book contains a mathematical background section in the firstchapter to outline some important pre-required mathematical issues.However, this section is restricted to materials related directly tothe contents of differential geometry of the book and hence the readerand user should not expect this mathematical background section to becomprehensive in any way. General mathematical knowledge, plus possibleconsultation of mathematical textbooks related to other disciplines ofmathematics when needed, should therefore be considered.

The book is furnished with an index in the end ofthe book as well as sets of exercises in the end of each chapter toprovide useful revisions and practice. To facilitate linking relatedconcepts and parts, and hence ensure better understanding of theprovided materials, cross referencing is used extensively throughoutthe book where these referrals are hyperlinked in the electronicversion of the book for the convenience of the ebook users. The bookalso contains a considerable number of graphic illustrations to helpthe readers and users to visualize the ideas and understand theabstract concepts.

The materials of differential geometry arestrongly interlinked and hence any text about the subject, like thepresent one, will face the problem of arranging the materials in anatural order to ensure gradual development of concepts. In this bookwe largely followed such a scheme. However, this is not always possibleand hence in some cases references are provided to materials in laterparts of the book for concepts needed in earlier parts. Nevertheless,in most cases brief definitions of the main concepts are provided inthe first chapter in anticipation of more detailed definitions andinvestigations in the subsequent chapters.

Regarding the preparation of the book, everythingis made by the author including all the graphic illustrations,indexing, typesetting, book cover, as well as overall design. In thisregard, I should acknowledge the use of LaTeX typesetting package andthe LaTeX based document preparation package LyX which facilitated thetypesetting and design of the book substantially.

Taha Sochi

London, March 2017

Table of Contents

Nomenclature

In the following table, we define some of thecommon symbols, notations and abbreviations which are used in the bookto provide easy access to the reader.

nabla differential operator
	Laplacian operator
~	isometric to
, subscript	partial derivative with respect to the following index(es)
; subscript	covariant derivative with respect to the following index(es)
1D, 2D, 3D, n D	one-, two-, three-, n -dimensional
overdot (e.g. r )	derivative with respect to general parameter t
prime (e.g. r )	derivative with respect to natural parameter s
t	absolute derivative with respect to t
, i	partial derivative with respect to th and ith variables
a	determinant of surface covariant metric tensor
a	surface covariant metric tensor
a11,a12,a22	coefficients of surface covariant metric tensor
a11,a12,a22	coefficients of surface contravariant metric tensor
a , a , a	surface metric tensor or its components
b	determinant of surface covariant curvature tensor
b	surface covariant curvature tensor
B	binormal unit vector of space curve
b11,b12,b22	coefficients of surface covariant curvature tensor
b , b , b	surface curvature tensor or its components
C	curve
CB , CN , CT	spherical indicatrices of curve C
Ce , Ci	evolute and involute curves
Cn	of class n
c , c , c	tensor of third fundamental form or its components
d	Darboux vector
d1 , d2	unit vectors in Darboux frame
det	determinant of matrix
ds	length of infinitesimal element of curve
dsB,dsN,dsT	length of line element in binormal, normal, tangent directions
d	area of infinitesimal element of surface
e,f,g	coefficients of second fundamental form
E,F,G	coefficients of first fundamental form
, , V	number of edges, faces and vertices of polyhedron
Ei , Ej	covariant and contravariant space basis vectors
E , E	covariant and contravariant surface basis vectors
Eq./Eqs.	Equation/Equations
f	function
Fig./Figs.	Figure/Figures
g	topological genus of closed surface
gij , gij	space metric tensor or its components
H	mean curvature
IS , IIS , IIIS	first, second and third fundamental forms
IS , IIS	tensors of first and second fundamental forms
iff	if and only if
J	Jacobian of transformation between two coordinate systems
J	Jacobian matrix
K	Gaussian curvature
Kt	total curvature
L	length of curve
n	normal unit vector to surface
N	principal normal unit vector to curve
P	point
r , R	radius
Ricci curvature scalar
r	position vector
r , r	1st and 2nd partial derivative of r with subscripted variables
R1 , R2	principal radii of curvature
n	n -dimensional space (usually Euclidean)
Rij , R ij	Ricci curvature tensor of 1st and 2nd kind for space
R , R	Ricci curvature tensor of 1st and 2nd kind for surface
Rijkl	Riemann-Christoffel curvature tensor of 1st kind for space
R	Riemann-Christoffel curvature tensor of 1st kind for surface

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