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Riemannian Geometry Andreas Kriegl Email:Andreas.Kriegl@Univie.Ac.At Riemannian Geometry Andreas Kriegl email:[email protected] 250067, WS 2018, Mo. 945-1030 and Th. 900-945 at SR9 z y x This is preliminary english version of the script for my homonymous lecture course in the Winter Semester 2018. It was translated from the german original using a pre and post processor (written by myself) for google translate. Due to the limitations of google translate { see the following article by Douglas Hofstadter www.theatlantic.com/. /551570 { heavy corrections by hand had to be done af- terwards. However, it is still a rather rough translation which I will try to improve during the semester. It consists of selected parts of the much more comprehensive differential geometry script (in german), which is also available as a PDF file under http://www.mat.univie.ac.at/∼kriegl/Skripten/diffgeom.pdf. In choosing the content, I followed the curricula, so the following topics should be considered: • Levi-Civita connection • Geodesics • Completeness • Hopf-Rhinov theoroem • Selected further topics from Riemannian geometry. Prerequisite is the lecture course 'Analysis on Manifolds'. The structure of the script is thus the following: Chapter I deals with isometries and conformal mappings as well as Riemann sur- faces - i.e. 2-dimensional Riemannian manifolds - and their relation to complex analysis. In Chapter II we look again at differential forms in the context of Riemannian manifolds, in particular the gradient, divergence, the Hodge star operator, and most importantly, the Laplace Beltrami operator. As a possible first application, a section on classical mechanics is included. In Chapter III we first develop the concept of curvature for plane curves and space curves, then for hypersurfaces, and finally for Riemannian manifolds. Of course, we will also treat geodesics, parallel transport and the covariant derivative. During the semester, I will post a detailed list of the treated sections in http://www.mat.univie.ac.at/∼kriegl/LVA-2018-WS.html. Of course, the attentive reader will be able to find (typing) errors. I kindly ask to let me know about them (consider the german saying: shared suffering is half the suffering). Future generations of students might appreciate it. Andreas Kriegl, Vienna in July 2018 Contents I. Conformal structures and Riemannian surfaces1 1. Conformal mappings1 2. Riemann surfaces6 3. Riemann mapping theorem and uniformization theorem9 II. Differential forms on Riemannian manifolds 11 4. Volume form and Hodge-Star operator 11 5. The Laplace Beltrami operator 13 6. Classical mechanics 26 III. Curvature und geodesics 37 7. Curvature of curves in the plane 37 8. Curvatures of curves in higher dimensions 44 9. Curvatures of hypersurfaces 48 10. Geodesics 67 11. Integral theorem of Gauß - Bonnet 76 12. Parallel transport 86 13. Covariant derivative 89 14. Curvatures of Riemannian manifolds 105 15. Jacobi Fields 121 16. The Cartan method of moving frames 132 Bibliography 143 Index 149 [email protected] c January 31, 2019 iii I. Conformal structures and Riemannian surfaces 1. Conformal mappings A Riemann metric on a smooth manifold M is a 2-fold covariant tensor field 2 1 ∗ ∗ ∼ 2 1 g 2 T (M) = C (M T M ⊗ T M) = L 1 (X(M); X(M); C (M; )); 0 C (M;R) R which is pointwise a positive-definite symmetric bilinear form, see [95, 24.1] and [95, 20.1]. It has therefore a representation with respect to local coordinates (ui) of the following form: X i j g = gi;j du ⊗ du : i;j A Riemannian manifold (M; g) is a smooth manifold M which is provided with an distinguished Riemann metric g, see [95, 18.11]. On Riemannian manifolds (M; g) we can define the length of tangential vectors ξx 2 TxM as jξxj := p gx(ξx; ξx) and, in analogy to [82, 6.5.12], the length of smooth curves c : [0; 1] ! M as Z 1 q 0 0 L(c) := gc(t)(c (t); c (t)) dt: 0 1.1 Definition (Parameterization according to arc length). A parameterization c of a curve is called parameterization by arc length if jc0(t)j = 1 for all t. For the length with respect to such parameterizations we thus b have La(c) = b − a by [87, 2.7]. 1.2 Proposition (Parameterization by arc length). Each curve has a parameterization by arc length. Each two such parameterizations of the same curve are equivalent via a parameter change of the form t 7−! ±t + a. Proof. Existence: Let c : I ! (M; g) be a curve (which we always assume to 0 t be regular, i.e. c (t) 6= 0 for all t), a a point in the interval I and s(t) := La(c) = R t 0 0 0 a jc (t)j dt the length function s : R ⊃ I ! R, with derivative s (t) = jc (t)j > 0. In particular, s(I) is connected and thus again an interval (see [81, 3.4.3]). The inverse function ' : s(I) ! I, s 7−! t(s) is smooth by the Inverse Function Theorem (see [81, 4.1.10] and [82, 6.3.5]). The parameterizationc ¯ := c ◦ ' is the required parameterization by arc length, since dc¯ dc dt dc 1 dc 1 dc 1 = · = · = · = · : ds dt ds dt ds dt jc0(t)j dt dc dt j dt j Uniqueness: If c and c ◦ ' are two parameterizations by arc length, then: 1 = j(c ◦ ')0(t)j = jc0('(t))j · j'0(t)j = j'0(t)j for all t; since j(c ◦ ')0(t)j = 1 = jc0('(t))j. Hence '0 = ±1 and thus '(t) = '(a) + R t 0 a ' (r)dr = '(a) ± (t − a) = ±t + ('(a) ∓ a). [email protected] c January 31, 2019 1 1.6 1. Conformal mappings + On connected Riemannian manifolds (M; g), we obtain a metric dg : M × M ! R in sense of topology n 1 o dg(p; q) := inf L(c): c 2 C (R;M); c(0) = p; c(1) = q : We have shown in [95, 18.12] that this metric dg generates the topology of M. 1.3 Definition (Isometry). If (M; g) and (N; h) are two Riemannian manifolds and f : M 7! N is smooth, then f is called isometry if and only if Txf :(TxM; gx) ! (Tf(x)N; hf(x)) is a linear isometry for all x (see [87, 1.2]). Note that f is an isometry if and only if f ∗h = g is. Remark. 1. If f is an isometry and c : R ! M is smooth, then Lh(f ◦ c) = Lf ∗h(c) = Lg(c): Thus we obtain dh(f(x); f(y)) ≤ df ∗h(x; y) = dg(x; y) for the distance, that is, the isometry can not increase the distance. If f is a diffeomorphism and an isometry then d(x; y) = d(f(x); f(y)). 2. If the set of fixed points of an isometry can be parameterized as a smooth curve c, then this curve is locally the shortest connection of each of two points: We will see in 10.8 that locally the shortest connections exist and are unique. But since the isometric image of such a curve has the same length, it must agree with it, that is, must be contained in the fixed point set. 1.4 Theorem of Nash. Each abstract and connected m-dimensional Riemannian manifold (M; g) can be isometrically embedded in R(2m+1)(6m+14). Without proof, see [125]. 1.5 Proposition (Existence of Riemannian metrics). Each paracompact smooth manifold admits a complete Riemannian metric, that is, a Riemannian metric g, whose associated metric dg on M is complete. Proof. We only need to embed (the connected components of) M into an Rn and then take the metric induced by the standard metric to obtain a Riemannian metric on M. Or we can use charts to find Riemannian metrics locally and glue them to a global Riemannian metric by using a partition of unity. This works, since \being a Rie- mann metric" is a convex condition. The existence of complete Riemannian metrics will be shown in 13.14 . 1.6 Proposition (Lie group of isometries). Let (M; g) be a connected m-dimensional Riemannian manifold, then Isom(M) := ff 2 Diff(M): f is an isometryg 1 can be made into a Lie group of dimension at most 2 m(m + 1). 2 [email protected] c January 31, 2019 1. Conformal mappings 1.9 The group Isom(M) is thus finite-dimensional in contrast to the group Diff(M) of all diffeomorphisms. For example, both Isom(Rm) = O(m) n Rm and Isom(Sm) = m(m−1) (m+1)(m+1−1) O(m + 1) have dimension 2 + m = 2 . Without proof. See [78, 2.1.2]. Since one can define angles between vectors by hx; yi cos ^(x; y) := phx; xiphy; yi by means of an inner product h ; i, we can measure angles α between tangent vectors on each Riemannian manifold (M; g) and thus between curves c1 and c2 in their intersection points (say c1(0) = c2(0)) in the following way: g(c0 (0); c0 (0)) cos α := 1 2 : p 0 0 p 0 0 g(c1(0); c1(0)) g(c2(0); c2(0)) 1.7 Definition (Conformal mappings). A smooth mapping f :(M; g) ! (N; h) is called angle preserving (or confor- mal) if Txf : TxM ! Tf(x)N is angle preserving for all x 2 M. 1.8 Theorem (Lie group of conformal diffeomorphisms). The set of conformal diffeomorphism of an m-dimensional paracompact connected 1 Riemannian manifold forms a Lie group of dimension at most 2 (m + 1)(m + 2).
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