Differential Geometry II Lecture 1: Introduction and Motivation

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1 Differential Geometry II Lecture 1: Introduction and Motivation Robert Haslhofer 1 Content of this lecture This course is on Riemannian geometry and the geometry of submanifol. The goal of this first lecture is to informally introduce and motivate some ideas, and to get an intuition for curvature. Therefore, this lecture is in a story telling format (starting in Lecture 2, we will give a more systematic treatment, with proper definitions, theorems, proofs, etc). 2 Riemannian manifol and submanifol A Riemannian manifold is a smooth manifold M equipped with a positive definite symmetric two-tensor g, called the Riemmanian metric. In essence, this means that at every point p M we have a symmetric positive definite matrix g ij (p) which allows us to compute the inner product of tangent vectors X, Y T p M: X, Y = n g ij (p)x i Y j. (2.1) i,j=1 In particular, we can measure lengths and angles: X = X, X, cos α = X, Y X Y. (2.2) Here are some examples of Riemannian manifol: Euclidean Space R n with the flat Euclidian metric g ij = δ ij. The Sphere S n with the metric induced from the embedding S n R n+1. Hyperbolic space H n = R n + with the metric g ij = δ ij. x 2 n Projective space CP n with the metric induced by S 1 S 2n+1 CP n. 1

2 More generally, whenever we have a submanifold M n R N of Euclidian space, then we get a metric g on M by restricting the ambient Euclidean metric to each tangent space T p M R N. By Whitney s embedding theorem any manifold can be embedded into some Euclidean space, and by Nash s embedding theorem any Riemannian manifold can be isometrically embedded into some Euclidian space. Thus, in principle it would suffice to talk about submanifol of Euclidian space, but it is often very helpful to think more abstractly in terms of (M, g). Every Riemannian manifold (M, g) is in particular a metric space (M, d). Indeed, if γ : I M is a piecewise smooth curve, then we can compute its length L(γ) = γ(t) dt, (2.3) and thus we can measure the distance I d(x, y) = inf{l(γ) γ is a piecewise smooth curve from x to y}. (2.4) Exercise 1. Convince yourself that d indeed satisfies the axioms of a metric. The infimum is always attained by a so-called minimizing geodesic. 3 A first glimpse at curvature A major part of this course will deal with curvature. Curvature has to do with second derivatives. To get some intuition, we consider curves and surfaces. Curvature of curves in R 2 Let x(s) be a smooth curve in R 2 parametrized by arclength, i.e. d x 1. The unit tangent vector is given by the first derivative t(s) = d x. The curvature vector is given by the second derivative k(s) = d2 x. Since we assumed that the 2 curve is parametrized by arclength, the unit tangent vector has indeed unit length. Differentiating the formula t(s), t(s) = 1, we get k(s), t(s) = 0, i.e. the curvature vector is perpendicular to the unit tangent vector. Thus, after choosing a unit normal vector n(s), we can write k(s) = k(s) n(s), and all information about the curvature is contained in the scalar quantity k(s). Exercise 2. Let k be the curvature of a curve γ in R 2. Show that: k = 1 R, where R is the radius of the osculating circle. k = dθ, where θ is the angle between the unit tangent vector and the x-axis. k = uxx, if we write γ as y = u(x). (1+u 2 x )3/2 By integration this implies: 2

3 Theorem 3.1 If γ is a closed curve in R 2, then γ k(s) = 2πw for some w = w(γ) Z. The number w is called the winding number, and equals ±1 for embedded curves. Curvature of curves in R 3 Let γ be a smooth closed curve in R 3. As before, if γ is parametrized by arclength, then the curvature vector is given by the second derivative k(s) = d2 γ, and it 2 can be shown that γ k(s) 2π. The following theorem shows a beautiful interaction between curvature and topology. Theorem 3.2 (Fary-Milnor) If γ is knotted curve in R 3, then γ k(s) > 4π. Curvature of surfaces in R 3 Let M 2 R 3 be a closed surface. At each point p M there are two principal curvatures k 1 and k 2, which are given by the minimal and maximal curvature of curves through p (with respect to the inward unit normal). We can then define the mean curvature H = k 1 + k 2, (3.1) 2 and the Gauss curvature, K = k 1 k 2. (3.2) E.g. for the unit sphere S 2 R 3 we have k 1 = k 2 = 1 and thus H = 1 and K = 1. The mean curvature is an extrinsic quantity, i.e. it depen on the embedding M 2 R 3 on not just on the intrinsic geometry of (M 2, g). In contrast we have: Theorem 3.3 (Gauss Theorema Egregium) The Gauss curvature K is intrinsic. As an illustrative example, we consider different isometric embeddings of a piece of paper into R 3. The value of the Gauss curvature is always zero, while the value of the mean curvature depen on the embedding. One way to prove Gauss remarkable theorem is to derive the formula A(r) = πr 2 π 12 K(p)r4 + o(r 4 ), (3.3) where A(r) is the area of the disc of intrinsic radius r about p. Since everything can be computed using only the metric g, this shows that K(p) is intrinsic. The formula 3

4 also illustrates the relationship between the sign of the curvature and volume growth. Continuing the beautiful interplay between curvature and topology, we have the classical Gauss-Bonnet theorem which relates the total curvature integral and the Euler-characteristic. Theorem 3.4 (Gauss-Bonnet) If (M 2, g) is a closed surface, then M K da = 2πχ(M). The other natural integrand to consider is the square of the mean curvature. Note that S 2 H 2 da = 4π. In 1965, it was conjectured by Willmore that in the case of the torus the integral is at least 2π 2, and indeed this was proved recently: Theorem 3.5 (Marques-Neves, 2012) If T 2 R 3 is an embedded torus, then T H2 da 2π 2. Equality is attained on the Clifford torus, aka the optimal doughnut. Curvature in higher dimensions If (M n, g) is a Riemannian manifold of general dimension, then for every two-plane σ T p M there is a sectional curvature K(σ). It is defined as the Gauss-curvature at p of the surface one gets by mapping σ to M via the exponential map. Again there is a beautiful interaction between curvature and topology. E.g. we will learn the proof of the following classical theorem. Theorem 3.6 (Cartan-Hadamard) If (M n, g) is a complete Riemannian manifold with K 0, then its universal cover is diffeomorphic to R n. About positive sectional curvature, let me mention the differential sphere theorem, which gives a topological conclusion if K min K max > 1 4 at every point. Theorem 3.7 (Brendle-Schoen, 2007) If (M n, g) has quarter-pinched positive sectional curvature, then M is diffeomorphic to S n. The constant 1 4 is sharp. Indeed CPn with its canonical metric has K min K max = Tentative plan for this course In the next few month we will systematically cover the theory of Riemannian manifol. We ll start with the definition and examples, and will show that every manifold 4

5 admits a Riemannian metric. After discussing some basic concepts like isometries, length and volume we will introduce the Levi-Civita connection, which allows us to compute derivatives of vector fiel, X Y. A key feature about calculus on manifol is that second derivatives don t commute any more. The difference is measured by the Riemann curvature tensor, i j X k j i X k = n R ijkl X l. (4.1) The Riemann curvature tensor contains all information about the sectional curvatures, and in particular the Ricci curvature and scalar curvature. We will learn many theorems about the interaction of curvature, geometry and topology, e.g. the Gauss-Bonnet and Cartan-Hadamard theorem mentioned above, but also some comparison theorems about Ricci curvature like Bishop-Gromov and Bonnet-Myers. A central point in many proofs is to investigate how the behavior of geodesics depen on the curvatures of the Riemannian manifold. I also plan to cover submanifol, the second fundamental form, minimal surfaces, and some of their applications (e.g. the Schoen-Yau proof of the positive mass theorem). l=1 5