IntroductiontoDifferentialGeometryofSpaceCurvesandSurfaces

Preface The present book is about differential geometry of space curves and surfaces. The formulation and presentation are largely based on a tensor calculus approach, which is the dominant trend in the modern mathematical literature of this subject, rather than the geometric approach which is usually found in some old style books. The book is prepared, to some extent, as part of tutorials about topics and applications related to tensor calculus. It can therefore 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. Apart from general background knowledge in a number of mathematical branches such as calculus, geometry and algebra, an important requirement for the reader and user of this book is familiarity with the terminology, notation and concepts of tensor calculus at reasonable level since many of the notations and concepts of differential geometry in its modern style are based on tensor calculus. The book contains a mathematical background section in the first chapter to outline some important pre-required mathematical issues. However, this section is restricted to materials related directly to the contents of differential geometry of the book and hence the reader and user should not expect this mathematical background section to be comprehensive in any way. General mathematical knowledge, plus possible consultation of mathematical textbooks related to other disciplines of mathematics when needed, should therefore be considered. The book is furnished with an index in the end of the book as well as sets of exercises in the end of each chapter to provide useful revisions and practice. To facilitate linking related concepts and parts, and hence ensure better understanding of the provided materials, cross referencing is used extensively throughout the book where these referrals are hyperlinked in the electronic version of the book for the convenience of the ebook users. The book also contains a considerable number of graphic illustrations to help the readers and users to visualize the ideas and understand the abstract concepts. The materials of differential geometry are strongly interlinked and hence any text about the subject, like the present one, will face the problem of arranging the materials in a natural order to ensure gradual development of concepts. In this book we largely followed such a scheme. However, this is not always possible and hence in some cases references are provided to materials in later parts of the book for concepts needed in earlier parts. Nevertheless, in most cases brief definitions of the main concepts are provided in the first chapter in anticipation of more detailed definitions and investigations in the subsequent chapters. Regarding the preparation of the book, everything is made by the author including all the graphic illustrations, indexing, typesetting, book cover, as well as overall design. In this regard, I should acknowledge the use of LaTeX typesetting package and the LaTeX based document preparation package LyX which facilitated the typesetting and design of the book substantially. Taha Sochi London, March 2017 Table of Contents Preface Nomenclature 1: Preliminaries 1.1: Differential Geometry 1.2: General Remarks, Conventions and Notations 1.3: Classifying the Properties of Curves and Surfaces 1.3.1: Local versus Global Properties 1.3.2: Intrinsic versus Extrinsic Properties 1.4: General Mathematical Background 1.4.1: Geometry and Topology 1.4.2: Functions 1.4.3: Coordinates, Transformations and Mappings 1.4.4: Intrinsic Distance 1.4.5: Basis Vectors 1.4.6: Flat and Curved Spaces 1.4.7: Homogeneous Coordinate Systems 1.4.8: Geodesic Coordinates 1.4.9: Christoffel Symbols for Curves and Surfaces 1.4.10: Riemann-Christoffel Curvature Tensor 1.4.11: Ricci Curvature Tensor and Scalar 1.5: Exercises 2: Curves in Space 2.1: General Background about Curves 2.2: Mathematical Description of Curves 2.3: Curvature and Torsion of Space Curves 2.3.1: Curvature 2.3.2: Torsion 2.4: Geodesic Torsion 2.5: Relationship between Curve Basis Vectors and their Derivatives 2.6: Osculating Circle and Sphere 2.7: Parallelism and Parallel Propagation 2.8: Exercises 3: Surfaces in Space 3.1: General Background about Surfaces 3.2: Mathematical Description of Surfaces 3.3: Surface Metric Tensor 3.3.1: Arc Length 3.3.2: Surface Area 3.3.3: Angle Between Two Surface Curves 3.4: Surface Curvature Tensor 3.5: First Fundamental Form 3.6: Second Fundamental Form 3.6.1: Dupin Indicatrix 3.7: Third Fundamental Form 3.8: Fundamental Forms 3.9: Relationship between Surface Basis Vectors and their Derivatives 3.9.1: Codazzi-Mainardi Equations 3.10: Sphere Mapping 3.11: Global Surface Theorems 3.12: Exercises 4: Curvature 4.1: Curvature Vector 4.2: Normal Curvature 4.2.1: Meusnier Theorem 4.3: Geodesic Curvature 4.4: Principal Curvatures and Directions 4.5: Gaussian Curvature 4.6: Mean Curvature 4.7: Theorema Egregium 4.8: Gauss-Bonnet Theorem 4.9: Local Shape of Surface 4.10: Umbilical Point 4.11: Exercises 5: Special Curves 5.1: Straight Line 5.2: Plane Curve 5.3: Involute and Evolute 5.4: Bertrand Curve 5.5: Spherical Indicatrix 5.6: Spherical Curve 5.7: Geodesic Curve 5.8: Line of Curvature 5.9: Asymptotic Line 5.10: Conjugate Direction 5.11: Exercises 6: Special Surfaces 6.1: Plane Surface 6.2: Quadratic Surface 6.3: Ruled Surface 6.4: Developable Surface 6.5: Isometric Surface 6.6: Tangent Surface 6.7: Minimal Surface 6.8: Exercises 7: Tensor Differentiation over Surfaces 7.1: Exercises References Author Notes 8: Footnotes Nomenclature In the following table, we define some of the common symbols, notations and abbreviations which are used in the book to provide easy access to the reader. nabla differential operator 2Laplacian operator isometric to , subscriptpartial derivative with respect to the following index(es) ; subscriptcovariant derivative with respect to the following index(es) 1D, 2D, 3D, n D one-, two-, three-, n -dimensional overdot (e.g. ) derivative with respect to general parameter t prime (e.g. r ) derivative with respect to natural parameter s tabsolute derivative with respect to t , i partial derivative with respect to th and i th variables adeterminant of surface covariant metric tensor asurface covariant metric tensor a 11 , a 12 , a 22coefficients of surface covariant metric tensor a 11 , a 12 , a 22coefficients of surface contravariant metric tensor a , a , a surface metric tensor or its components bdeterminant of surface covariant curvature tensor bsurface covariant curvature tensor Bbinormal unit vector of space curve b 11 , b 12 , b 22coefficients of surface covariant curvature tensor b , b , b surface curvature tensor or its components Ccurve C B , C N , C Tspherical indicatrices of curve C C e , C ievolute and involute curves C nof class n c , c , c tensor of third fundamental form or its components dDarboux vector d 1 , d 2unit vectors in Darboux frame detdeterminant of matrix dslength of infinitesimal element of curve ds B , ds N , ds T length of line element in binormal, normal, tangent directions d area of infinitesimal element of surface e , f , gcoefficients of second fundamental form E , F , Gcoefficients of first fundamental form , , Vnumber of edges, faces and vertices of polyhedron E i , E jcovariant and contravariant space basis vectors E , E covariant and contravariant surface basis vectors Eq./Eqs.Equation/Equations ffunction Fig./Figs.Figure/Figures gtopological genus of closed surface g ij , g ijspace metric tensor or its components Hmean curvature I S , II S , III Sfirst, second and third fundamental forms I S , II Stensors of first and second fundamental forms iffif and only if JJacobian of transformation between two coordinate systems JJacobian matrix KGaussian curvature K ttotal curvature Llength of curve nnormal unit vector to surface Nprincipal normal unit vector to curve Ppoint r , Rradius Ricci curvature scalar rposition vector r , r 1 st and 2 nd partial derivative of r with subscripted variables R 1 , R 2principal radii of curvature nn -dimensional space (usually Euclidean) R ij , R i jRicci curvature tensor of 1 st and 2 nd kind for space R , R Ricci curvature tensor of 1 st and 2 nd kind for surface R ijkl Riemann-Christoffel curvature tensor of 1 st kind for space R Riemann-Christoffel curvature tensor of 1 st kind for surface R i jklRiemann-Christoffel curvature tensor of 2 nd kind for space R Riemann-Christoffel curvature tensor of 2 nd kind for surface R radius of curvature R radius of torsion snatural parameter of curve representing arc length Ssurface S Ttangent surface of space curve tgeneral parameter of curve Tfunction period Ttangent unit vector of space curve T P Stangent space of surface S at point P trtrace of matrix ugeodesic normal vector u 1 , u 2surface coordinates u surface coordinate u , vsurface coordinates x ispace coordinate x i surface basis vector in full tensor notation x , y , zcoordinates in 3D space (usually Cartesian) ij , k Christoffel symbol of 1 st kind for space , Christoffel symbol of 1 st kind for surface k ijChristoffel symbol of 2 nd kind for space Christoffel symbol of 2 nd kind for surface ij , ij , j icovariant, contravariant and mixed Kronecker delta ij klgeneralized Kronecker delta discriminant of quadratic equation i 1 i n , i 1 i n covariant, contravariant relative permutation tensor in n D space i 1 i n , i 1 i n covariant, contravariant absolute permutation tensor in n D space angle or parameter ssum of interior angles of polygon curvature of curve 1 , 2principal curvatures of surface at a given point B , Tcurvature of binormal and tangent spherical indicatrices g , ngeodesic and normal curvatures gu , gvgeodesic curvatures of u and v coordinate curves nu , nvnormal curvatures of u and v coordinate curves Kcurvature vector K g , K ngeodesic and normal components of curvature vector direction parameter of surface real parameter pseudo-radius of pseudo-sphere , polar coordinates of plane , , zcylindrical coordinates of 3D space area of surface patch torsion of curve B , Ttorsion of binormal and tangent spherical indicatrices ggeodesic torsion angle or parameter Euler characteristic real parameter Note : due to the restrictions on the availability and visibility of symbols in the mobi format, as well as similar formatting issues, we should draw the attention of the ebook readers to the following points: 1. Bars over symbols, which are used in the printed version, were replaced by tildes. However, for convenience we kept using the terms “barred” and “unbarred” in the text to refer to the symbols with and without tildes. 2. The square root symbol in mobi is ( ) where the argument is contained inside the parentheses. For example, the square root of a is symbolized as ( a ) . 3. In the mobi format, superscripts are automatically displayed before subscripts unless certain measures are taken to force the opposite which may distort the look of the symbol and may not even be the required format when the superscripts and subscripts should be side by side which is not possible in the mobi text and live equations. Therefore, for convenience and aesthetic reasons we only forced the required order of the subscripts and superscripts or used imaged symbols when it is necessary; otherwise we left the symbols to be displayed according to the mobi choice although this may not be ideal like displaying the Christoffel symbol of the second kind as: or the generalized Kronecker delta as: instead of their normal look as: and . Chapter 1 Preliminaries In this chapter, we provide preliminary materials in the form of a general introduction about differential geometry, remarks about the conventions and notations used in this book, classification of the properties of curves and surfaces, and a general mathematical background related to differential geometry of curves and surfaces. 1.1 Differential Geometry Differential geometry is a branch of mathematics that largely employs methods and techniques of other branches of mathematics such as differential and integral calculus, topology and tensor analysis to investigate geometrical issues related to abstract objects, such as space curves and surfaces, and their properties where these investigations are mostly focused on these properties at small scales. The investigations of differential geometry also include characterizing categories of these objects. There is also a close link between differential geometry and the disciplines of differential topology and differential equations. Differential geometry may be contrasted with “algebraic geometry” which is another branch of geometry that uses algebraic tools to investigate geometric issues mainly of global nature. The investigation of the properties of curves and surfaces in differential geometry are closely linked. For instance, investigating the characteristics of space curves is extensively exploited in the investigation of surfaces since common properties of surfaces are defined and quantified in terms of the properties of curves embedded in these surfaces. For example, several aspects of the surface curvature at a point are defined and quantified in terms of the parameters of the surface curves passing through that point. 1.2 General Remarks, Conventions and Notations First, we should remark that the present book is largely based on investigating curves and surfaces embedded in a 3D flat space coordinated by a rectangular Cartesian system. In most cases, “surface” and “space” in the present book mean 2D and 3D manifolds respectively. Another remark is that twisted curves can reside in a 2D manifold (surface) or in a higher dimensionality manifold (usually 3D space). Hence we usually use “surface curves” and “space curves” to refer to the type of the manifold of residence. However, in most cases a single curve can be viewed as resident of more than one manifold and hence it is a surface and space curve at the same time. For example, a curve embedded in a surface which in its turn is embedded in a 3D space is a surface curve and a space curve at the same time. Consequently, in this book these terms should be interpreted flexibly. Many statements formulated in terms of a particular type of manifold can be correctly and easily extended to another type with minimal adjustments of dimensionality and symbolism. Moreover, “space” in some statements should be understood in its general meaning as a manifold embracing the curve not as opposite to “surface” and hence it can include a 2D space, i.e. surface. Following the convention of several authors, when discussing issues related to 2D and 3D manifolds the Greek indices range over 1, 2 while the Latin indices range over 1, 2, 3 . Therefore, the use of Greek and Latin indices should in general indicate the type of the intended manifold unless it is stated otherwise. We use u indexed with superscript Greek letters (e.g. u and u ) to symbolize surface coordinates (see Footnote 1 in 8 ) while we largely use x indexed with superscript Latin letters (e.g. x i and x j ) to represent space Cartesian coordinates although they are sometimes used to represent space general curvilinear coordinates. Comments are usually added to clarify the situation if necessary. A related issue is that the indexed E are mostly used for the surface, rather than the space, basis vectors where they are subscripted or superscripted with Greek indices, e.g. E and E . However, in some cases indexed E are used for space basis vectors; in which case they are distinguished by using Latin indices, e.g. E i and E j . When the basis vectors are indexed numerically rather than symbolically, the distinction between surface and space bases should be obvious from the context if it is not stated explicitly. Regarding the Christoffel symbols of the first and second kind of various manifolds, they may be based on the space metric or the surface metric. Hence, when a number of Christoffel symbols in a certain context or equation are based on more than one metric, the type of indices, i.e. Greek or Latin, can be used as an indicator to the underlying metric where the Greek indices represent the surface (e.g. , and ) while the Latin indices represent the space (e.g. ij , k and k ij ). Nevertheless, comments are generally added to account for potential oversight. In particular, when the Christoffel symbols are numbered (e.g. 1 22 ), instead of being indexed symbolically, comments will be added to clarify the situation. For brevity, convenience and clean notation in certain contexts, we use an overdot (e.g. ) to indicate derivative with respect to a general parameter t while we use a prime (e.g. r ) to indicate derivative with respect to a natural parameter s representing arc length. For the same reasons, subscripts are also used occasionally to symbolize partial derivative with respect to the subscript variables, e.g. f v and r . We should also remark that we follow the summation convention, which is largely used in tensor calculus. Comments are added in a few exceptional cases where this convention does not apply. We deliberately use a variety of notations for the same concepts for the purpose of convenience and to familiarize the reader with different notations all of which are in common use in the literature of differential geometry and tensor calculus. Having proficiency in these subjects requires familiarity with these various, and sometimes conflicting, notations. Moreover, in some situations the use of one of these notations or the other is either necessary or advantageous depending on the location and context. An important example of using di