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Riemannian Geometry Riemannian Geometry Contents Chapter 1. Smooth manifolds 5 1. Tangent vectors, cotangent vectors and tensors 5 2. The tangent bundle of a smooth manifold 5 3. Vector fields, covector fields, tensor fields, n-forms 5 Chapter 2. Riemannian manifolds 7 1. Riemannian metric 7 2. The three model geometries 9 3. Connections 13 4. Geodesics and parallel translation along curves 16 5. The Riemannian connection 17 6. Connections on submanifolds and pull-back connections 19 7. Geodesics in the three geometries 20 8. The exponential map and normal coordinates 21 9. The Riemann distance function 25 Chapter 3. Curvature 29 1. The Riemann curvature tensor 29 2. Ricci curvature, scalar curvature, and Einstein metrics 31 3. Riemannian submanifolds 33 4. Sectional curvature 36 5. Jacobi fields 38 6. Comparison theorems 44 Chapter 4. Space-times 47 Chapter 5. Multilinear Algebra 49 1. Tensors 49 2. Tensors of inner product spaces 51 3. Coordinate expressions 52 Chapter 6. Non-euclidean geometry 55 1. The hyperbolic plane 55 Bibliography 59 3 CHAPTER 1 Smooth manifolds 1. Tangent vectors, cotangent vectors and tensors 1.1. Lemma. Let F : M m → N n be a smooth map. Suppose that (x1, . , xm) are local co- ordinates on M and (y1, . , yn) local coordinates on N. Then ∂ ∂(yiF ) ∂ (1.2) F∗( ∂xj ) = ∂xj ∂yi , 1 ≤ j ≤ m, ∗ i ∂(yiF ) j i (1.3) F (dy ) = ∂xj dx = d(y F ), 1 ≤ i ≤ n t ∂(yiF ) ∂(yiF ) The (n × m)-matrix ∂xj is the matrix for F∗ and the (m × n)-matrix ∂xj is the matrix for F ∗. In other words, ∂ ∂ ∂ ∂ ∂(yiF ) F∗ ∂x1 ...F∗ ∂xm = ∂y1 ... ∂yn ∂xj F ∗dy1 dx1 i . = ∂(y F ) . . ∂xj . F ∗dyn dxm ∗ i ∂ i ∂ ∂ i ∂yiF Proof. F (dy )( ∂xj ) = dy (F∗( ∂xj )) = F∗ ∂xj (y ) = ∂xj . 2. The tangent bundle of a smooth manifold 3. Vector fields, covector fields, tensor fields, n-forms 1.4. Proposition. The differential d: C∞(M) → T 1(M) is an R-linear map satisfying the derivational rules du udv vdu − udv d(uv) = (du)v + u(dv), d u = − = v v v2 v2 for all smooth functions u, v ∈ C∞(M) (where v(p) 6= 0 for all p ∈ M in the last formula). If F : M → N is a smooth map, then the diagram F ∗ C∞(M) o C∞(N) d d F ∗ T 1(M) o T 1(N) commutes meaning that F ∗(du) = d(uF ) for all u ∈ C∞(N). n+1 2 Pn+1 i 2 1.5. Example. Let s: R → R be the smooth map s(x) = |x| = i=1 (x ) . Then s−1(1) = Sn ⊂ Rn is the sphere of radius R. The differential ds = P 2xidxi is the linear n+1 n+1 P i i ⊥ map R = TpR → R given by dsp(v) = 2p v = 2hp, vi with kernel ker dsp = p at any n n n+1 n+1 point p 6= 0. Let p be any point of S and ι∗ : TpS → TpR = R the linear map induced n by the inclusion, ι. For any tangent vector X ∈ TpS , ds(ι∗X) = X(sι) = X(1) = 0. Hence the tangent space at p is the kernel of dsp, n ⊥ n+1 n+1 TpS = p ⊂ R = TpR and the tangent bundle of Sn, TSn = {(p, v) ⊂ Sn × Rn+1 | hv, pi = 0} ⊂ Sn × Rn+1 5 6 1. SMOOTH manifoldS is the vector bundle whose fibre over any p ∈ Sn is p⊥. A smooth vector field on Sn is a smooth map v : Sn → Rn+1 such that v(p) ⊥ p for all p ∈ Sn. Show that any odd sphere has a vector field without zeros. Does S2 admit a smooth vector field with no zeros? Can you describe T RP n? Can you describe TM if M = f −1(0) consists of the manifold solutions to the equation f(x) = 0 for some smooth map f : Rn+1 → R? k 1.6. Definition. A smooth -tensor field on M is a smooth section of the tensor bundle ` k T` (M) → M. Particular cases are 0 ∞ •T0 (M) = C (M) 0 •T1 (M) consists of vector fields on M 1 •T0 (M) consists of 1-forms on M Tensor fields admit ⊗ •T k1 (M) × T k1 (M) −→T k1+k2 (M) (tensor product of tensor fields) `1 `1 `1+`2 k+1 tr k •T`+1 (M) −→T` (M) (contraction of tensor fields) 1 0 1.7. Example. Let ω ∈ T0 (M) be a 1-form and X ∈ T1 (M) a vector field on M. Then 1 X ⊗ ω ∈ T 1(M) is -tensor field with contraction tr(X ⊗ ω) = ω(X) (5.7). 1 1 k In a coordinate patch any -tensor field A is (5.5) a C∞(M)-linear combination ` (1.8) A = Aj1···j` ∂ ⊗ · · · ∂ ⊗ dxi1 ⊗ · · · dxik i1···ik j1 j` of tensor products of the basis tensor fields and basis 1-forms. The smooth functions Aj1···j` are i1···ik called the components of the tensor field A. ∗ P∞ k The tensor algebra of M is the graded algebra T (M) = k=0 T (M) equipped with the tensor product T r(M) × T s(M) −→T⊗ r+s(M). If F : M → N is a smooth map, F ∗ : T k(N) → T k(M) ∗ k is the linear map given by F (A)(X1,...,Xk) = A(F∗X1,...,F∗Xk) for all A ∈ T (N) and all smooth vector fields X1,...,Xk on M. 1.9. Lemma. T ∗(M) is a graded algebra. F ∗ : T ∗(N) → T ∗(M) is a homomorphism of C∞(N)- algebras: F ∗(aω ⊗ η) = F ∗(a)F ∗(ω) ⊗ F ∗(η). CHAPTER 2 Riemannian manifolds Riemann’s idea was that in the infinitely small, on a scale much smaller than the the smallest particle, we do not know if Euclidean geometry is still in force. Therefore we better not assume that this is the case and instead open up for the possibility that in the infinitely small there may be other length functions, there may be other inner products on the tangent space! A Riemannian manifold is a smooth manifold equipped with inner product, which may or may not be the Euclidean inner product, on each tangent space. 1. Riemannian metric 2.1. Definition. A Riemannian metric on a smooth manifold M is a symmetric, positive 2 definite -tensor g ∈ T 2(M). 0 0 In a coordinate frame we may write i j g = gijdx ⊗ dx , gij = g(∂i, ∂j) i j i j This means that g(U ∂i,V ∂j) = gijU V and in particular that (2.1) h∂i, ∂ii = gii, h∂i, ∂ji = gij 1 Note that there are only 2 n(n + 1) different functions here as gij = gji by symmetry. 2.2. Remark. Since the metric tensor is symmetric, it is traditional to write it in a basis of symmetric tensors. The symmetrization of ω ⊗ η is the tensor 1 ωη = (ω ⊗ η + η ⊗ ω) 2 Note that ωη = ηω and that ω2 = ωω = ω ⊗ ω. Observe that n i j X i 2 X i j g = gijdx dx = 2 gii(dx ) + 2 gijdx dx i=1 1≤i<j≤n 2.3. Lemma. Let F : M → N be an immersion and g a Riemannian metric on N. (1) F ∗g is a Riemannian metric on M. i j (2) If g = gijdy dy in a coordinate frame on N, then ∗ −1 i j F (g)|F (U) = gijdy F dy F Proof. It is a general fact that F ∗(g) is a smooth 2-form on M (1.9). F ∗(g) is symmetric because g is symmetric and it is positive definite because g is positive definite and F∗ is injective ∗ i j (1.9) ∗ i ∗ j (1.4) i j on each fibre. F (gijdy dy ) = gijF dy F dy = gijdy F dy F . For instance if F : U → M is a parameterization (an inverse chart) of an open subset of M ⊂ Rm, then the pull-back of the induced metric on M is m ∗ ∗ i j i j X i 2 (2.4) F (g) = F (δijdx dx ) = δijdF dF = (dF ) i=1 These expressions are tensor fields living in the tensor algebra T ∗(M) of M. 7 8 2. RIEMANNIAN MANIFOLDS 2.5. Example. (Graphs) Let M ⊂ Rn × R be the graph of the smooth function f : M → R. Then s(x) = (x, f(x)) is a diffeomorphism so that the Riemannian manifold (M, ι∗gn+1) is isometric to (Rn, s∗gn+1) where the metric s∗gn+1 is n+1 n n X X ∂f 2 X ∂f ∂f s∗( (dxi)2) = (dxi)2 + dxi = (dxi)2 + dxidxj ∂xi ∂xi ∂xj i=1 i=1 i=1 n ∂f ∂f X ∂f X ∂f ∂f = δ + dxidxj = 1 + ( )2(dxi)2 + 2 dxidxj ij ∂xi ∂xj ∂xi ∂xi ∂xj i=1 1≤i<j≤n 2 2 3 2.6. Example. let S+ be the upper hemisphere on S ⊂ R considered as the graph of the p 2 2 2 2 2 ∗ 3 function f(x, y) = 1 − x − y defined on the unit ball B ⊂ R . Then (S+, ι g ) is isometric to (B2, s∗g3) where !2 −x −y s∗g3 = (dx)2 + (dy)2 + dx + dy p1 − x2 − y2 p1 − x2 − y2 1 − y2 1 − x2 2xy = (dx)2 + (dy)2 + dxdy 1 − x2 − y2 1 − x2 − y2 1 − x2 − y2 This means (2.1) that 1 − y2 1 − x2 xy h∂ , ∂ i = , h∂ , ∂ i = , h∂ , ∂ i = x x 1 − x2 − y2 y y 1 − x2 − y2 x y 1 − x2 − y2 2 at the point (x, y) ∈ B .
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