DIFFERENTIAL GEOMETRY, D COURSE Gabriel P. Paternain Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge CB3 0WB, England E-mail address: [email protected] Contents Chapter 1. Smooth manifolds and smooth maps 1 1.1. Definitions 1 1.2. Tangent spaces and derivatives 2 1.3. Regular values and Sard’s theorem 3 1.4. Transversality 5 1.5. Manifolds with boundary 6 1.6. Degree modulo 2 7 1.7. Abstract manifolds and Whitney’s embedding theorem 11 Chapter 2. Length, area and curvature 13 2.1. Arc-length, curvature and torsion of curves 13 2.2. The isoperimetric inequality in the plane 14 2.3. First fundamental form and Area 16 2.4. The Gauss map 19 2.5. The second fundamental form in local coordinates 20 2.6. Theorema Egregium 21 Chapter 3. Critical points of length and area 23 3.1. Geodesics 23 3.2. Covariant derivative and parallel transport 24 3.3. Minimal surfaces 25 3.4. The Weierstrass representation 26 3.5. The meaning of g in the Weierstrass representation 29 Chapter 4. Global Riemannian geometry 31 4.1. The exponential map and geodesic polar coordinates 31 4.2. The Gauss-Bonnet theorem 33 4.3. Applications 38 4.4. Fenchel’s theorem 39 4.5. The F´ary-Milnor theorem 40 3 CHAPTER 1 Smooth manifolds and smooth maps 1.1. Definitions Definition 1.1. Let U be an open set of Rn. A map f : U → Rm is called smooth if it has continuous partial derivatives of all orders. A map f : X → Rm defined on an arbitrary subset X of Rn is called smooth if for each x ∈ X there is an open set U ⊂ Rn containing x and a smooth map F : U → Rm such that F equals f on U ∩ X. We will use the shorthand term local when we wish to refer to behaviour only in a neighbourhood of a point. Smoothness is a local property. (The term global refers to the whole space X.) Definition 1.2. A smooth map f : X → Y between subsets of Euclidean space is a diffeomorphism if it is a bijection, and if the inverse f −1 : Y → X is also smooth. X and Y are diffeomorphic if such an f exists. Differential Topology is about properties of a set X ⊂ Rn which are invariant under diffeomorphisms. Arbitrary sets of Rn may be too wild. We would like to have a class of sets on which can do locally the same as in Euclidean space. Definition 1.3. Let X be a subset of RN . We say that X is a k-dimensional manifold if each point possesses a neighbourhood V in X which is diffeomorphic to an open set of Rk. A diffeomorphism φ : U → V , where U is an open set of Rk, is called a parametrization of the neighbourhood V . The inverse diffeomorphism φ−1 : V → U is called a coordinate system or a chart on V . −1 If we write φ = (x1, . , xk) the k smooth functions x1, . , xk on V are called coordinate functions. The dimension k of X is written as dim X. Example 1.4. The unit sphere S2 given by all (x, y, z) ∈ R3 with x2 +y2 +z2 = 1 is a smooth manifold of dimension 2. The diffeomorphism p (x, y) 7→ (x, y, 1 − x2 − y2), for x2 +y2 < 1, parametrizes the region z > 0 of S2. By interchanging the roles of x, y and z and changing the sign of the variables, we obtain similar parametrizations of the regions x > 0, y > 0, x < 0, y < 0 and z < 0. Since these cover S2, it follows that S2 is a smooth manifold. Similarly Sn := {x ∈ Rn+1 : |x| = 1} is a smooth manifold of dimension n. Exercise 1.5. If X and Y are manifolds, so is X × Y , and dim X × Y = dim X + dim Y . 1 2 1. SMOOTH MANIFOLDS AND SMOOTH MAPS Definition 1.6. If X and Z are both manifolds in RN and Z ⊂ X, then Z is a submanifold of X. In particular, X itself is a submanifold of RN . Any open set of X is a submanifold of X. The codimension of Z in X is dim X − dim Z. 1.2. Tangent spaces and derivatives n For any open set U ⊂ R and x ∈ U, the tangent space to U at x (denoted TxU) is defined to be Rn. Recall from Analysis II that for any smooth map f : U → Rm, the derivative of f at x ∈ U is the linear map n m dfx : R → R given by f(x + th) − f(x) dfx(h) = lim t→0 t for h ∈ Rn. Now let us define the tangent space TxX of an arbitrary smooth manifold X ⊂ RN . Choose a parametrization φ : U → X around x where U is an open set of Rk. Without loss of generality assume that φ(0) = x. Think of φ as a map from U to N k N R , so dφ0 : R → R is defined. Set k TxX := dφ0(R ). Lemma 1.7. TxX does not depend on φ and dim TxX = k. Proof. Suppose ψ : V → X is another choice, with ψ(0) = x. By shrinking both U and V , we may assume that φ(U) = ψ(V ). Then the map h = ψ−1 ◦ φ : U → V is a diffeomorphism. Now we write φ = ψ ◦ h and we differentiate. Using the chain rule we have: dφ0 = dψ0 ◦ dh0. Since dh0 is an invertible linear map, it follows at once that dφ0 and dψ0 have the same image. Since φ−1 : φ(U) → U is a smooth map, we can choose an open set W in RN containing x and a smooth map Φ : W → Rk that extends φ−1. Then Φ ◦ φ is the identity map of U, so the chain rule implies that the composition of linear maps k dφ0 dΦx k R → TxX → R k k is the identity map of R . Thus dφ0 : R → TxX is an isomorphism and dim TxX = k. We can now define the derivative of a smooth map f : X → Y between arbitrary manifolds. Let x be a point in X and set y = f(x). The derivative must be a linear map dfx : TxX → TyY which gives us back the usual derivative when X and Y are open sets in Euclidean space and it must also satisfy the chain rule. Keeping this in mind, let φ : U → X and ψ : V → Y be parametrizations around x and y respectively, where U is an open set of Rk and V is an open set of Rl. As before suppose φ(0) = x and ψ(0) = y. If U is small enough we have the diagram: f X −−−−→ Y x x φ ψ h=ψ−1◦f◦φ U −−−−−−−−→ V 1.3. REGULAR VALUES AND SARD’S THEOREM 3 Hence our definition of dfx must be so that the following diagram commutes: dfx TxX −−−−→ TyY x x dφ0 dψ0 dh0 Rk −−−−→ Rl Since dφ0 is an isomorphism, we must have −1 dfx := dψ0 ◦ dh0 ◦ dφ0 . We must verify that this definition does not depend on the choices of φ and ψ. Exercise 1.8. Prove that this is indeed the case. f g Chain Rule. If X → Y → Z are smooth maps of manifolds, then d(g ◦ f)x = dgf(x) ◦ dfx. 1.2.1. The inverse function theorem. Let f : X → Y be a smooth map between manifolds. We say that f is a local diffeomorphism at x if f maps a neighbourhood of x diffeomorphically onto a neighbourhood of f(x). Theorem 1.9 (The inverse function theorem). Suppose that f : X → Y is a smooth map whose derivative dfx at the point x is an isomorphism. Then f is a local diffeomorphism at x. Exercise 1.10. Prove the theorem assuming the inverse function theorem for smooth functions between open sets of Euclidean space, as you have seen it in Analysis II. 1.3. Regular values and Sard’s theorem Let f : X → Y be a smooth map between manifolds. Let C be the set of all points x ∈ X such that dfx : TxX → Tf(x)Y is not surjective. Definition 1.11. A point in C will be called a critical point. A point in f(C) will be called a critical value. A point in the complement of f(C) will be called a regular value. Remark 1.12. Note that if dim X < dim Y , then C = X and the preimage of a regular value is the empty set. Theorem 1.13 (Preimage theorem). Let y be a regular value of f : X → Y with dim X ≥ dim Y . Then the set f −1(y) is a submanifold of X with dim f −1(y) = dim X − dim Y . −1 Proof. Let x ∈ f (y). Since y is a regular value, the derivative dfx maps TxX onto TyY . The kernel of dfx is a subspace K of TxX of dimension p := dim X − dim Y . Suppose X ⊂ RN and let T : RN → Rp be any linear map such that Ker(T ) ∩ K = {0}. Consider the map F : X → Y × Rp given by F (z) = (f(z),T (z)). The derivative of F is given by dFx(v) = (dfx(v),T (v)) 4 1. SMOOTH MANIFOLDS AND SMOOTH MAPS which is clearly nonsingular by our choice of T . By the inverse function theorem, F is a local diffeomorphism at x, i.e.