9.2. THEORY OF SURFACES 395 9.2.3. Intrinsic Geometry of Surface 9.2.3-1. Intrinsic geometry and bending of surface. Suppose that two surfaces U and U ∗ are given and there is one-to-one correspondence between their points such that the length of each curve on U is equal to the length of the corresponding curve on U ∗ . Such a one-to-one mapping of U into U ∗ is called a bending of the surface U into the surface U ∗ , and the surfaces U and U ∗ are said to be applicable. The correspondence between U and U ∗ is said to be isometric. The intrinsic geometry of a surface studies geometric constructions and quantities related to the surface that can be determined solely from the first quadratic form. The notions of length of a segment, angle between two curves, and area of part of a surface all belong in intrinsic geometry. On the opposite, the curvature of a curve given on the surface by the equations u = u(t), v = v(t) cannot be found using only the first quadratic form, and hence it does not belong in intrinsic geometry. 9.2.3-2. Index notation. Surface as Riemannian space. From now on in this chapter, the following notation related to tensor analysis is used: u 1 = u, u 2 = v; g 11 = E, g 12 = g 21 = F , g 22 = G; b 11 = L, b 12 = b 21 = M , b 22 = N ; r 1 = r u , r 2 = r v , r 12 = r 21 = r vu = r uv , r 11 = r uu , r 22 = r vv . In the new notation, the first fundamental quadratic form becomes Edu 2 + 2Fdudv+ Gdv 2 = 2 α=1 2 β=1 g αβ du α du β = g αβ du α du β , and the second fundamental quadratic form is Ldu 2 + 2Mdudv+ Ndv 2 = 2 α=1 2 β=1 b αβ du α du β = b αβ du α du β . The expressions for the coefficients of the first and second quadratic forms in the new notation become g ij = r i ⋅ r j , b ij = r ij ⋅ N, where N is the unit normal vector to the surface; i and j are equal to either 1 or 2. 396 DIFFERENTIAL GEOMETRY 9.2.3-3. Derivation formula. The Christoffel symbols Γ k,ij of the first kind are defined to be the scalar products of the vectors r k and r ij ; i.e., r k ⋅ r ij = Γ k,ij . The Christoffel symbols of the first kind satisfy the formula Γ k,ij = 1 2 ∂g ki ∂u j + ∂g kj ∂u i – ∂g ij ∂u k . This is one of the basic formulas in the theory of surfaces; this formula means that the scalar products of the second partial derivatives of the position vector r(u i , u j ) by its partial derivatives can be expressed in terms of the coefficients of the fi rst quadratic form (more precisely, in terms of their derivatives). The Christoffel symbols Γ k ij of the second kind are defined by the relations Γ k ij = n t=1 g kt Γ k,ij , where g kt is given by n k=1 g kt g ks = 1,ift = s, 0,ift ≠ s. The Christoffel symbols of the second kind are the coefficients in the decomposition of the vector r ij in two noncollinear vectors r 1 and r 2 and the unit normal vector N: r ij = Γ 1 ij r 1 + Γ 2 ij r 2 + b ij N (i, j = 1, 2). (9.2.3.1) Formulas (9.2.3.1) are called the first group of derivation formulas (the Gauss derivation formulas). The formulas N 1 =–b 1 1 r 1 – b 2 1 r 2 , N 1 =–b 1 2 r 1 – b 2 2 r 2 ,(9.2.3.2) where b j i =b iα g αj , arecalled the second group of derivation formulas (Weingarten formulas). The formulas in the second group of derivation formulas express the partial derivatives of the unit normal vector N in terms of the variables u 1 and u 2 in the decomposition in the basis vectors r 1 , r 2 ,andN. Formulas (9.2.3.1) and (9.2.3.2) express the partial derivatives with respect to u 1 and u 2 of the basis vectors, i.e., of the two tangent vectors r 1 and r 2 and the normal vector N, at a given point on a surface. These partial derivatives of r 1 , r 2 and N are obtained as a decomposition in the vectors r 1 , r 2 and N themselves. 9.2.3-4. Gauss formulas. Peterson–Codazzi formulas. If the first quadratic form of the surface is given, then the second quadratic form cannot be chosen arbitrarily, since its discriminant (LN –M 2 ) is completely determined by the Gauss formula b 11 b 22 – b 2 12 = ∂ 2 g 12 ∂u 1 ∂u 2 – 1 2 ∂ 2 g 11 ∂u 2 ∂u 2 – 1 2 ∂ 2 g 22 ∂u 1 ∂u 1 + Γ γ 12 Γ δ 12 g γδ – Γ α 11 Γ β 22 g αβ ,(9.2.3.3) where γ, δ, α,andβ are independent summation indices equal to either 1 or 2. REFERENCES FOR CHAPTER 9 397 We use (9.2.3.3) to reduce relation (9.2.2.2) to the form K(u, v)=k 1 k 2 = LN – M 2 EG – F 2 = b 11 b 22 – b 2 12 g 11 g 22 – g 2 12 .(9.2.3.4) In view of (9.2.3.4), the Gaussian curvature of the surface is completely determined by the coefficients of the first quadratic form and by their first and second derivatives with respect to u 1 and u 2 . The coefficients b 11 , b 12 , b 22 of the second quadratic form and their first derivatives are related to the coefficients g 11 , g 12 , g 22 of the first quadratic form and their first derivatives (contained only in Γ k ij ) by differential equations known as the Peterson–Codazzi formulas: ∂b i1 ∂u 2 – Γ 1 i2 b 11 – Γ 2 i2 b 21 = ∂b i2 ∂u 1 – Γ 1 i1 b 12 – Γ 2 i1 b 22 (i = 1, 2). The Gauss formula and the Peterson–Codazzi formulas are necessary and sufficient conditions for two analytically determined quadratic differential forms to be the fist and second quadratic forms of some surface. References for Chapter 9 Aubin, T., A Course in Differential Geometry, American Mathematical Society, Providendce, Rhoad Island, 2000. Burke, W. L., Applied Differential Geometry, Cambridge University Press, Cambridge, 1985. Byushgens, S. S., Differential Geometry [in Russian], Komkniga, Moscow, 2006. Chern, S S., Chen, W H., and Lam, K. S., Lectures on Differential Geometry, World Scientific Publishing Co., Hackensack, New Jersey, 2000. Danielson, D. A., Vectors and Tensors in Engineering and Physics, 2nd Rep Edition, Westview Press, Boulder, Colorado, 2003. Dillen, F. J. E. and Verstraelen, L. C. A., Handbook of Differential Geometry, Vol. 1, North Holland, Amsterdam, 2000. Dillen, F. J. E. and Verstraelen, L. C. A., Handbook of Differential Geometry, Vol. 2, North Holland, Amsterdam, 2006. Guggenheimer, H. W., Differential Geometry, Dover Publications, New York, 1977. Kay, D. C., Schaum’s Outline of Tensor Calculus, McGraw-Hill, New York, 1988. Kobayashi, S. and Nomizu, K., Foundations of Differential Geometry, Vol. 1, Wiley, New York, 1996. Kreyszig, E., Differential Geometry, Dover Publications, New York, 1991. Lang, S., Fundamentals of Differential Geometry, Springer, New York, 2001. Lebedev, L. P. and Cloud, M. J., Tensor Analysis, World Scientific Publishing Co., Hackensack, New Jersey, 2003. Lovelock, D. and Rund, H., Tensors, Differential Forms, and Variational Principles, Dover Publications, New York, 1989. O’Neill, V., Elementary Differential Geometry, Rev. 2nd Edition, Academic Press, New York, 2006. Oprea, J., Differential Geometry and Its Applications, 2nd Edition, Prentice Hall, Englewood Cliffs, New Jersey, 2003. Pogorelov, A. V., Differential Geometry, P. Noordhoff, Groningen, 1967. Postnikov, M. M.,Linear Algebra and Differential Geometry (Lectures in Geometry), Mir Publishers, Moscow, 1982. Pressley, A., Elementary Differential Geometry, Springer, New York, 2002. Rashevsky, P. K., A Course in Differential Geometry, 4th Edition [in Russian], URSS, Moscow, 2003. Simmonds,J.G.,A Brief on Tensor Analysis, 2nd Edition, Springer, New York, 1997. Somasundaram, D., Differential Geometry: A First Course, Alpha Science International, Oxford, 2004. Spain, B., Tensor Calculus: A Concise Course, Dover Publications, New York, 2003. Spivak, M., A Comprehensive Introduction to Differential Geometry. Vols 1–5, 3rd Edition, Publish or Perish, Houston, 1999. Struik, D. J., Lectures on Classical Differential Geometry, 2nd Edition, Dover Publications, New York, 1988. Temple, G., Cartesian Tensors: An Introduction, Dover Publications, New York, 2004. Chapter 10 Functions of Complex Variable 10.1. Basic Notions 10.1.1. Complex Numbers. Functions of Complex Variable 10.1.1-1. Complex numbers. The set of complex numbers is an extension of the set of real numbers. An expression of the form z = x + iy,wherex and y are real numbers, is called a complex number,andthe symbol i is called the imaginary unit: i 2 =–1. The numbers x and y are called, respectively, the real and imaginary parts of z and denoted by x =Rez and y =Imz.(10.1.1.1) The complex number x + i0 is identified with real number x, and the number 0 + iy is denoted by iy and is said to be pure imaginary. Two complex numbers z 1 = x 1 + iy 1 and z 2 = x 2 + iy 2 are assumed to be equal if x 1 = x 2 and y 1 = y 2 . The complex number ¯z = x – iy is said to be conjugate to the number z. The sum or difference of complex numbers z 1 = x 1 + iy 1 and z 2 = x 2 + iy 2 is defined to be the number z 1 z 2 = x 1 x 2 + i(y 1 y 2 ). (10.1.1.2) Addition laws: 1. z 1 + z 2 = z 2 + z 1 (commutativity). 2. z 1 +(z 2 + z 3 )=(z 1 + z 2 )+z 3 (associativity). The product z 1 z 2 of complex numbers z 1 = x 1 + iy 1 and z 2 = x 2 + iy 2 is definedtobe the number z 1 z 2 =(x 1 x 2 – y 1 y 2 )+i(x 1 y 2 – x 2 y 1 ). (10.1.1.3) Multiplication laws: 1. z 1 z 2 = z 2 z 1 (commutativity). 2. z 1 (z 2 z 3 )=(z 1 z 2 )z 3 (associativity). 3. (z 1 + z 2 )z 3 = z 1 z 3 + z 2 z 3 (distributivity with respect to addition). The product of a complex number z = x + iy by its conjugate is always nonnegative: z¯z = x 2 + y 2 .(10.1.1.4) For a positive integer n,then-fold product of z by itself is called the nth power of the number z and is denoted by z n . A number w is called an nth root of a number z and is denoted by w = n √ z if w n = z. If z 2 ≠ 0, then the quotient of z 1 and z 2 is defined as z 1 z 2 = x 1 x 2 + y 1 y 2 x 2 2 + y 2 2 + i x 2 y 1 – x 1 y 2 x 2 2 + y 2 2 .(10.1.1.5) Relation (10.1.1.5) can be obtained by multiplying the numerator and the denominator of the fraction z 1 /z 2 by ¯z 2 . 399 400 FUNCTIONS OF COMPLEX VARIABLE 10.1.1-2. Geometric interpretation of complex number. There is a one-to-one correspondence between complex numbers z = x + iy and points M with coordinates (x, y) on the plane with a Cartesian rectangular coordinate system OXY or with vectors −−→ OM connecting the origin O with M (Fig. 10.1). The length r of the vector −−→ OM is called the modulus of the number z and is denoted by r = |z|, and the angle ϕ formed by the vector −−→ OM and the positive direction of the OX-axis is called the argument of the number z and is denoted by ϕ =Argz. X Y r φ M O Figure 10.1. Geometric interpretation of complex number. The modulus of a complex number is determined by the formula |z| = √ z¯z = x 2 + y 2 .(10.1.1.6) The argument Arg z is determined up to a multiple of 2π,Argz =argz + 2kπ,wherek is an arbitrary integer and arg z is the principal value of Arg z determined by the condition –π <argz ≤ π. The principal value arg z is given by the formula arg z = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ arctan(y/x)forx > 0, π +arctan(y/x)forx < 0, y ≥ 0, –π +arctan(y/x)forx < 0, y < 0, π/2 for x = 0, y > 0, –π/2 for x = 0, y < 0. (10.1.1.7) For z = 0,Argz is undefined. Since x = r cos ϕ and y = r sin ϕ, it follows that the complex number can be written in the trigonometric form z = x + iy = r(cos ϕ + i sin ϕ). (10.1.1.8) For numbers z 1 = r 1 (cos ϕ 1 +isin ϕ 1 )andz 2 = r 2 (cos ϕ 2 +isin ϕ 2 ), written in trigonometric form, the following rules of algebraic operations are valid: z 1 z 2 = r 1 r 2 cos(ϕ 1 + ϕ 2 )+i sin(ϕ 1 + ϕ 2 ) , z 1 z 2 = r 1 r 2 cos(ϕ 1 – ϕ 2 )+i sin(ϕ 1 – ϕ 2 ) . (10.1.1.9) In the latter formula, it is assumed that z ≠ 0. For any positive integer n,thisimpliesthede Moivre formula z n = r n (cos nϕ + i sin nϕ), (10.1.1.10) as well as the formula for extracting the root of a complex number.Forz ≠ 0,thereare exactly n distinct values of the nth root of the number z = r(cos ϕ + i sin ϕ). They are determined by the formulas n √ z = n √ r cos ϕ + 2kπ n + i sin ϕ + 2kπ n (k = 0, 1, 2, , n – 1). (10.1.1.11) 10.1. BASIC NOTIONS 401 Example. Let us find all values of 3 √ i. We represent the complex number z = i in trigonometric form. We have r = 1 and ϕ =argz = 1 2 π. The distinct values of the third root are calculated by the formula ω k = 3 √ i cos π 2 + 2πk 3 + i sin π 2 + 2πk 3 (k = 0, 1, 2), so that ω 0 =cos π 6 + i sin π 6 = √ 3 2 + i 1 2 , ω 1 =cos 5π 6 + i sin 5π 6 =– √ 3 2 + i 1 2 , ω 2 =cos 3π 2 + i sin 3π 2 =–i. The roots are shown in (Fig. 10.2). X Y i i ω ω ω 1 2 0 √3 √3 2 22 Figure 10.2. The roots of 3 √ i. X Y O z z+z zz z 1 1 1 2 2 2 Figure 10.3. The sum and difference of complex numbers. The plane OXY is called the complex plane,theaxisOX is called the real axis,and the axis OY is called the imaginary axis. The notions of complex number and point on the complex plane are identical. The geometric meaning of the operations of addition and subtraction of complex num- bers is as follows: the sum and the difference of complex numbers z 1 and z 2 are the vectors equal to the directed diagonals of the parallelogram spanned by the vectors z 1 and z 2 (Fig. 10.3). The following inequalities hold (Fig. 10.3): |z 1 + z 2 | ≤ |z 1 | + |z 2 |, |z 1 – z 2 | ≥ |z 1 | – |z 2 | .(10.1.1.12) Inequalities (10.1.1.12) become equalities if and only if the arguments of the complex numbers z 1 and z 2 coincide (i.e., arg z 1 =argz 2 ) or one of the numbers is zero. 10.1.2. Functions of Complex Variable 10.1.2-1. Notion of function of complex variable. A subset D of the complex plane such that each point of D has a neighborhood contained in D (i.e., D is open) and two arbitrary points of D can be connected by a broken line lying in D (i.e., D is connected) is called a domain on the complex plane. A point that does not itself lie in D but whose arbitrary neighborhood contains points of D is called a boundary point of D. The set of all boundary points of D is called the boundary of D, and the union . derivation formulas (Weingarten formulas). The formulas in the second group of derivation formulas express the partial derivatives of the unit normal vector N in terms of the variables u 1 and u 2 in. vectors r 1 , r 2 ,andN. Formulas (9.2.3.1) and (9.2.3.2) express the partial derivatives with respect to u 1 and u 2 of the basis vectors, i.e., of the two tangent vectors r 1 and r 2 and the normal. 2003. Dillen, F. J. E. and Verstraelen, L. C. A., Handbook of Differential Geometry, Vol. 1, North Holland, Amsterdam, 2000. Dillen, F. J. E. and Verstraelen, L. C. A., Handbook of Differential Geometry,