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DIFFERENTIAL GEOMETRY Philippe G. Ciarlet City University of Hong

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DIFFERENTIAL GEOMETRY Philippe G. Ciarlet City University of Hong AN INTRODUCTION TO DIFFERENTIAL GEOMETRY Philippe G. Ciarlet City University of Hong Kong Lecture Notes Series Contents Preface iii 1 Three-dimensional differential geometry 5 1.1 Curvilinear coordinates . 5 1.2 Metric tensor . 6 1.3 Volumes, areas, and lengths in curvilinear coordinates . 9 1.4 Covariant derivatives of a vector field and Christoffel symbols . 11 1.5 Necessary conditions satisfied by the metric tensor; the Riemann curvature tensor . 16 1.6 Existence of an immersion defined on an open set in R3 with a prescribed metric tensor . 18 1.7 Uniqueness up to isometries of immersions with the same metric tensor . 29 1.8 Continuity of an immersion as a function of its metric tensor . 36 2 Differential geometry of surfaces 53 2.1 Curvilinear coordinates on a surface . 53 2.2 First fundamental form . 57 2.3 Areas and lengths on a surface . 59 2.4 Second fundamental form; curvature on a surface . 60 2.5 Principal curvatures; Gaussian curvature . 66 2.6 Covariant derivatives of a vector field and Christoffel symbols on a surface; the Gauß and Weingarten formulas . 71 2.7 Necessary conditions satisfied by the first and second fundamen- tal forms: the Gauß and Codazzi-Mainardi equations; Gauß’ theorema egregium . 74 2.8 Existence of a surface with prescribed first and second fundamen- tal forms . 77 2.9 Uniqueness up to isometries of surfaces with the same fundamen- tal forms . 87 2.10 Continuity of a surface as a function of its fundamental forms . 92 References 101 i PREFACE The notes presented here are based on lectures delivered over the years by the author at the Universit´e Pierre et Marie Curie, Paris, at the University of Stuttgart, and at City University of Hong Kong. Their aim is to give a thorough introduction to the basic theorems of Differential Geometry. In the first chapter, we review the basic notions arising when a three- dimensional open set is equipped with curvilinear coordinates, such as the metric tensor, Christoffel symbols, and covariant derivatives. We then prove that the vanishing of the Riemann curvature tensor is sufficient for the existence of iso- metric immersions from a simply-connected open subset of Rn equipped with a Riemannian metric into a Euclidean space of the same dimension. We also prove the corresponding uniqueness theorem, also called rigidity theorem. In the second chapter, we study basic notions about surfaces, such as their two fundamental forms, the Gaussian curvature, Christoffel symbols, and co- variant derivatives. We then prove the fundamental theorem of surface theory, which asserts that the Gauß and Codazzi-Mainardi equations constitute suffi- cient conditions for two matrix fields defined in a simply-connected open subset of R3 to be the two fundamental forms of a surface in a three-dimensional Eu- clidean space. We also prove the corresponding rigidity theorem. In addition to such \classical" theorems, we also include in both chapters very recent results, which have not yet appeared in book form, such as the continuity of a surface as a function of its fundamental forms. The treatment is essentially self-contained and proofs are complete. The prerequisites essentially consist in a working knowledge of basic notions of anal- ysis and functional analysis, such as differential calculus, integration theory and Sobolev spaces, and some familiarity with ordinary and partial differential equations. These notes use some excerpts from Chapters 1 and 2 of my book \Mathe- matical Elasticity, Volume III: Theory of Shells", published in 2000 by North- Holland, Amsterdam; in this respect, I am indebted to Arjen Sevenster for his kind permission to reproduce these excerpts. Otherwise, the major part of these notes was written during the fall of 2004 at City University of Hong Kong; this part of the work was substantially supported by a grant from the Research Grants Council of Hong Kong Special Administrative Region, China [Project No. 9040869, CityU 100803]. Hong Kong, January 2005 iii Chapter 1 THREE-DIMENSIONAL DIFFERENTIAL GEOMETRY 1.1 CURVILINEAR COORDINATES To begin with, we list some notations and conventions that will be consistently used throughout. All spaces, matrices, etc., considered here are real. Latin indices and exponents vary in the set 1; 2; 3 , except when they are used for indexing sequences, and the summationf congvention with respect to repeated indices or exponents is systematically used in conjunction with this rule. For instance, the relation j gi(x) = gij (x)g (x) means that 3 j gi(x) = gij (x)g (x) for i = 1; 2; 3: j=1 X j ij Kronecker's symbols are designated by δi ; δij , or δ according to the context. Let E3 denote a three-dimensional Euclidean space, let a b and a b denote the Euclidean inner product and exterior product of a; b · E3, and^let a = pa a denote the Euclidean norm of a E3. The space 2E3 is endowedjwithj · 2 i an orthonormal basis consisting of three vectors e = ei. Let xi denote the Cartesian coordinates of a point x E3 and let @ := @=@x . 2 i i In addition, let there be given a three-dimensionalb bvector spbace in which i R3 three vectors e = ei form a basis.b This space wilbl be identifieb d with . Let xi 3 2 denote the coordinates of a point x R and let @i := @=@xi, @ij := @ =@xi@xj , 3 2 and @ijk := @ =@xi@xj @xk. Let there be given an open subset Ω of E3 and assume that there exist an open subset Ω of R3 and an injective mapping Θ : Ω E3 such that Θ(Ω) = Ω. ! Then each point x Ω can be unambiguouslyb written as 2 b x = Θ(x); x Ω; b b 2 b 5 6 Three-dimensional differential geometry [Ch. 1 and the three coordinates xi of x are called the curvilinear coordinates of x (Figure 1.1-1). Naturally, there are infinitely many ways of defining curvilinear coordinates in a given open set Ω, depending on how the open set Ω and theb mapping Θ are chosen! b g3(x) Ω Ωb Θ g2(x) x3 x^ x g1(x) e3 e^3 E3 R3 e2 e2 x2 ^ 1 e e^1 x1 Figure 1.1-1: Curvilinear coordinates and covariant bases in an open set Ω ⊂ E3. The three coordinates x1; x2; x3 of x 2 Ω are the curvilinear coordinates of x = Θ(xb) 2 Ω. If the three vectors gi(x) = @iΘ(x) are linearly independent, they form the cobvariant basisb at x = Θ(x) and they are tangent to the coordinate lines passing through x. b b Examples of curvilinear coordinates include the well-known cylindrical and spherical coordinates (Figure 1.1-2). In a different, but equally important, approach, an open subset Ω of R3 together with a mapping Θ : Ω E3 are instead a priori given. ! If Θ 0(Ω; E3) and Θ is injective, the set Ω := Θ(Ω) is open by the in- variance 2ofCdomain theorem (for a proof, see, e.g., Nirenberg [1974, Corollary 2, p. 17] or Zeidler [1986, Section 16.4]), and curvilinearb coordinates inside Ω are unambiguously defined in this case. If Θ 1(Ω; E3) and the three vectors @ Θ(x) are linearly independent bat all 2 C i x Ω, the set Ω is again open (for a proof, see, e.g., Schwartz [1992] or Zeidler [1986,2 Section 16.4]), but curvilinear coordinates may be defined only locally in this case: Givenb x Ω, all that can be asserted (by the local inversion theorem) is the existence of 2an open neighborhood V of x in Ω such that the restriction of Θ to V is a 1-diffeomorphism, hence an injection, of V onto Θ(V ). C 1.2 METRIC TENSOR Let Ω be an open subset of R3 and let Θ = Θ ei : Ω E3 i ! b Sect. 1.2] Metric tensor 7 Ω^ Ω^ x^ x^ z r E3 E3 ' ρ ' Figure 1.1-2: Two familiar examples of curvilinear coordinates. Let the mapping Θ be defined by Θ : ('; ρ, z) 2 Ω ! (ρ cos '; ρ sin '; z) 2 E3. Then ('; ρ, z) are the cylindrical coordinates of x = Θ('; ρ, z). Note that (' + 2kπ; ρ, z) or (' + π + 2kπ; −ρ, z); k 2 Z, are also cylindrical coordinates of the same point x and that ' is 3 b not defined if x is the origin of E . b Let the mappingb Θ be defined by Θ : ('; ; r) 2 Ω ! (r cos cos '; r cos sin '; r sin ) 2 E3: Then ('; ; r) are the spherical coordinates of x = Θ('; ; r). Note that (' + 2kπ; + 2`π; r) or (' + 2kπ; + π + 2`π; −r) are also spherical coordinates of the same point x and that ' 3 b and are not defined if x is the origin of E . b b be a mapping that is differentiable at a point x Ω. If δx is such that (x+δx) Ω, then 2 2 Θ(x + δx) = Θ(x) + rΘ(x)δx + o(δx); where the 3 3 matrix rΘ(x) is defined by × @1Θ1 @2Θ1 @3Θ1 rΘ(x) := @ Θ @ Θ @ Θ (x): 0 1 2 2 2 3 21 @1Θ3 @2Θ3 @3Θ3 @ A Let the three vectors g (x) R3 be defined by i 2 @iΘ1 g (x) := @ Θ(x) = @ Θ (x); i i 0 i 21 @iΘ3 @ A r i i.e., gi(x) is the i-th column vector of the matrix Θ(x) and let δx = δx ei. Then the expansion of Θ about x may be also written as i Θ(x + δx) = Θ(x) + δx gi(x) + o(δx): If in particular δx is of the form δx = δtei, where δt R and ei is one of the basis vectors in R3, this relation reduces to 2 Θ(x + δtei) = Θ(x) + δtgi(x) + o(δt): 8 Three-dimensional differential geometry [Ch.
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