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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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    Lecture 35: Hadamaard’s Theorem and Jacobi Fields Last time I stated: Theorem 35.1. Let (M,g) be a complete Riemannian manifold. Assume that for every p M and for every x, y T M, we have ∈ ∈ p K(x, y) 0. ≤ Then for any p M,theexponentialmap ∈ exp : T M M p p → is a covering map. 1. Jacobi fields—surfaces from families of tangent vectors Last time I also asked you to consider the following set-up: Let w : ( ,) TpM be a family of tangent vectors. Then for every tangent vector w−(s), we→ can consider the geodesic at p with tangent vector w(s). This defines amap f :( ,) (a, b) M, (s, t) γ (t)=exp(tw(s)). − × → → w(s) p Fixing s =0,notethatf(0,t)=γ(t)isageodesic,andthereisavectorfield ∂ f ∂s (0,t) which is a section of γ TM—e.g., a smooth map ∗ J :(a, b) γ∗TM. → Proposition 35.2. Let ∂ J(t)= f. ∂s (0,t) Then J satisfies the differential equation J =Ω(˙γ(t),J(t))γ ˙ (t). ∇∂t ∇∂t 117 Chit-chat 35.3. As a consequence, the acceleration of J is determined com- pletely by the curvature’s affect on the velocity of the geodesic and the value of J. Definition 35.4 (Jacobi fields). Let γ be a geodesic. Then any vector field along γ satisfying the above differential equation is called a Jacobi field. Remark 35.5. When confronted with an expression like Y or ∇X ∇X ∇Y there are only two ways to swap the order of the differentiation: Using the fact that one has a torsion-free connection, so Y = X +[X, Y ] ∇X ∇Y or by using the definition of the curvature tensor: = + +Ω(X, Y ).
  • Riemann's Contribution to Differential Geometry

    Riemann's Contribution to Differential Geometry

    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Historia Mathematics 9 (1982) l-18 RIEMANN'S CONTRIBUTION TO DIFFERENTIAL GEOMETRY BY ESTHER PORTNOY UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN, URBANA, IL 61801 SUMMARIES In order to make a reasonable assessment of the significance of Riemann's role in the history of dif- ferential geometry, not unduly influenced by his rep- utation as a great mathematician, we must examine the contents of his geometric writings and consider the response of other mathematicians in the years immedi- ately following their publication. Pour juger adkquatement le role de Riemann dans le developpement de la geometric differentielle sans etre influence outre mesure par sa reputation de trks grand mathematicien, nous devons &udier le contenu de ses travaux en geometric et prendre en consideration les reactions des autres mathematiciens au tours de trois an&es qui suivirent leur publication. Urn Riemann's Einfluss auf die Entwicklung der Differentialgeometrie richtig einzuschZtzen, ohne sich von seinem Ruf als bedeutender Mathematiker iiberm;issig beeindrucken zu lassen, ist es notwendig den Inhalt seiner geometrischen Schriften und die Haltung zeitgen&sischer Mathematiker unmittelbar nach ihrer Verijffentlichung zu untersuchen. On June 10, 1854, Georg Friedrich Bernhard Riemann read his probationary lecture, "iber die Hypothesen welche der Geometrie zu Grunde liegen," before the Philosophical Faculty at Gdttingen ill. His biographer, Dedekind [1892, 5491, reported that Riemann had worked hard to make the lecture understandable to nonmathematicians in the audience, and that the result was a masterpiece of presentation, in which the ideas were set forth clearly without the aid of analytic techniques.
  • Chapter 13 Curvature in Riemannian Manifolds

    Chapter 13 Curvature in Riemannian Manifolds

    Chapter 13 Curvature in Riemannian Manifolds 13.1 The Curvature Tensor If (M, , )isaRiemannianmanifoldand is a connection on M (that is, a connection on TM−), we− saw in Section 11.2 (Proposition 11.8)∇ that the curvature induced by is given by ∇ R(X, Y )= , ∇X ◦∇Y −∇Y ◦∇X −∇[X,Y ] for all X, Y X(M), with R(X, Y ) Γ( om(TM,TM)) = Hom (Γ(TM), Γ(TM)). ∈ ∈ H ∼ C∞(M) Since sections of the tangent bundle are vector fields (Γ(TM)=X(M)), R defines a map R: X(M) X(M) X(M) X(M), × × −→ and, as we observed just after stating Proposition 11.8, R(X, Y )Z is C∞(M)-linear in X, Y, Z and skew-symmetric in X and Y .ItfollowsthatR defines a (1, 3)-tensor, also denoted R, with R : T M T M T M T M. p p × p × p −→ p Experience shows that it is useful to consider the (0, 4)-tensor, also denoted R,givenby R (x, y, z, w)= R (x, y)z,w p p p as well as the expression R(x, y, y, x), which, for an orthonormal pair, of vectors (x, y), is known as the sectional curvature, K(x, y). This last expression brings up a dilemma regarding the choice for the sign of R. With our present choice, the sectional curvature, K(x, y), is given by K(x, y)=R(x, y, y, x)but many authors define K as K(x, y)=R(x, y, x, y). Since R(x, y)isskew-symmetricinx, y, the latter choice corresponds to using R(x, y)insteadofR(x, y), that is, to define R(X, Y ) by − R(X, Y )= + .