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Differential Geometry University of Cambridge Mathematics Tripos Part III Differential Geometry Michaelmas, 2018 Lectures by A. Kovalev Notes by Qiangru Kuang Contents Contents 1 Manifolds 2 2 Matrix Lie groups 6 3 Tangent space to manifolds 9 3.1 Lie algebra .............................. 10 3.2 Left-invariant vector field ...................... 15 4 Submanifolds 18 5 Differential forms 22 5.1 Orientation of manifolds ....................... 24 5.2 Exterior derivative .......................... 25 5.3 Pullback of differential forms .................... 26 5.4 de Rhan cohomology ......................... 27 5.5 Basic integration on manifolds ................... 29 6 Vector bundles 31 6.1 Structure group and transition functions .............. 32 6.2 Principal bundles ........................... 33 6.3 Hopf bundle .............................. 35 6.4 Pullback of a vector bundle or a principal bundle ......... 36 6.5 Bundle morphism ........................... 37 7 Connections 39 7.1 Curvature ............................... 44 8 Riemannian geometry 46 8.1 Geodesics ............................... 49 8.2 Curvature of a Riemannian metrix ................. 52 1 1 Manifolds 1 Manifolds We want to generalise curves and surfaces in R2. A curve is a map 훾 ∶ R ⊇ 퐼 → R2 or R3 that satisfies certain properties we’ll find out in a minute. Clearly continuity is necessary but not sufficent, as evidenced by the famous Peano space-filling curve. Smoothness is not quite enough either, as 푡 ↦ (푡2, 푡3) has a cusp at the origin. The correct requirement will be that 훾 has regular parame- terisation, i.e. | ̇훾(푡)| ≠ 0 for all 푡. Similarly, a surface should be defined as as a map 푟 ∶ R2 ⊇ 퐷 → R3 with a ∞ 휕푟 휕푟 regular parameterisation, i.e. 푟 ∈ 퐶 (퐷) and ∣ 휕푢 × 휕푣 ∣ ≠ 0 for all (푢, 푣) ∈ 퐷. We may follow this route and generalise to (hyper)surfaces in R푛, which will be a generalisation of classical differential geometry of curves and surfaces in R3. The good thing is that we can readily apply calculus and it is easy to construct these objects. However, it does suffer from the prolblem of different parameteri- sations give rise to different geometric objects, as well as some surface requiring more than one parameterisation. Although these can be bypassed more or less, the more serious drawback is the extra technical complexity determined by the higher dimension of the ambient space. A better concept is smooth manifolds. We first begin with a review of topo- logical structure, which every manifold possesses. Definition (topological space). A topological space 푀 is a choice of class of the open sets such that 1. ∅ and 푀 are open, 2. if 푈 and 푈 ′ are open then so is 푈 ∩ 푈 ′, 3. for anly collection of open sets, the union is open. In this course, we always require a topological space to be Hausdorff and second countable. Definition (local coordinate chart). A local coordinate chart on a topolog- ical space 푀 is a homeomorphism 휑 ∶ 푈 → 푉 where 푈 ⊆ 푀 and 푉 ⊆ R푑 are open. 푈 is a coordinate neighbourhood. Definition (퐶∞-differentiable structure). A 퐶∞-differentiable structure on a topological space 푀 is a collection of charts {휑훼 ∶ 푈훼 → 푉훼} where 푑 푉훼 ⊆ R for all 훼 such that 1. {푈훼} covers 푀, i.e. 푀 = ⋃ 푈훼, −1 ∞ 2. compatibility condition: for all 훼, 훽, 휑훽 ∘ 훽훼 is 퐶 wherever defined, i.e. on 휑훼(푈훼 ∩ 푈훽), 3. maximality: if 휑 is compatible with all the 휑훼’s then 휑 is in the collection. The collection of charts is called an atlas. −1 Note that 2 implies that 휑훽 ∘ 휑훼 is a diffeomorphism. 2 1 Manifolds Definition (manifold). A manifold is a Hausdorff, second countable topo- logical space with a 퐶∞-differentiable structure. Remark. 1. In practice, we almost never specify the topological structure. Instead, we can induce a topology from a 퐶∞ structure by declaring 퐷 ⊆ 푀 open if 푑 and only if for all (휑훼, 푈훼), 휑훼(푈훼 ∩ 퐷) is open in R . 2. We may replace 퐶∞ by 퐶푘 for 푘 > 0 finite. If we set 푘 = 0, the objects be- come topological manifolds. On the other hand, use C푛 and holomorphic maps we get complex manifolds. 3. Requirement of being Hausdorff and second countable are for rather tech- nical reasons. In some cases we may drop Hausdorffness requirement, and such examples do arise naturally. However in that case we lose uniqueness of limits. Similarly non-second countable space, such as the disjoint union of uncountably many R푛, can become manifold-like structure if we relax the definition. Example. 1. R푑 covered by the single chart 휑 = id. 푆푛 = {퐱 = (푥 , … , 푥 ) ∈ 푛+1 ∶ ∑푛 푥2 = 1} 2. Unit sphere 0 푛 R 푖=0 푖 . The charts are stereographic projections 1 휑(퐱) = (푥1, … , 푥푛) (1) 1 − 푥0 1 휓(퐱) = (푥1, … , 푥푛) (2) 1 + 푥0 where 휑 is defined for all points on 푆푛 except (1, 0, … , 0) and similar for 휓. Suppose 휑(푃 ) = 푢, 휓(푃 ) = 푣, then by basic geometry 푣 = 휓 ∘ 휑−1(푢) = 푢 ‖푢‖2 . 3. Given 푀1, 푀2 manifolds of dimension of 푑1 and 푑2, 푀1 ×푀2 is a manifold 1 1 of dimension 푑1 + 푑2. We define the 푛-torus to be 푇푛 =⏟⏟⏟⏟⏟ 푆 × ⋯ × 푆 . 푛 4. An open subset 푈 of a manifold 푀 is a manifold. 5. The real projective space, R푃 푛 = {all straight lines through 0 in R푛+1}. The points in the space are 푥0 ∶ ⋯ ∶ 푥푛 where 푥푖’s are not all zero. Note that 푥0 ∶ ⋯ ∶ 푥푛 = 휆푥0 ∶ … 휆푥푛 for all 휆 ≠ 0. As per a previous remark, we shall induce the topology by ∞ a 퐶 structure. The charts are (푈푖, 휑푖) where 푈푖 = {푥푖 ≠ 0}, and 푥0 ̂ 푥푛 휑푖(푥0 ∶ ⋯ ∶ 푥푛) = ( ,…, 푖, … , ) 푥푖 푥푖 3 1 Manifolds ̂ 푛 where 푖 denotes that the 푖th coordinate is omitted. The 푈푖’s cover R푃 so we are left to check compatibility. For 푖 < 푗, 푦 1 푦 −1 1 ̂ 푛 휑푗 ∘ 휑푖 ∶ (푦1, … , 푦푛) ↦ 푦1 ∶ … , 1⏟ ∶ 푦푛 ↦ ( ,…, ,…, 푗, … , ) 푖th 푦푗 푦푗 푦푗 is smooth. Since 푖 and 푗 are arbitrary, R푝푛 is a 푛-manifold. Similarly we can check C푃 푛 is a 2푛-manifold, and H푃 푛 is a 4푛-manifold (but this is a bit tricky due to noncommutativity). 6. Grassmannians (over R or C): we define Gr(푘, 푛) to be all 푘-dimensional subspaces of the 푛-dimensional vector space, which generalises the projec- tive space. For real vector spaces for example, R푃 푛 = Gr(1, 푛 + 1). We can check Gr(푘, 푛) is a manifold of dimension 푘(푛 − 푘). The construction is a bit technical so we will give example of one chart. Let 푈 be the 푘-subspaces obtainable as the span of rows of 푘×푛 matrices of the form (퐼푘 ∗), and the local coordinate maps a basis of such 푘-subspace to the 푘 × (푛 − 푘) block ∗. Since the frist 푘 rows are linearly independent, we call 푈 = 푈1<2<⋯<푘. More generally the domain of charts take the 푈 퐶∞ form 1≤푖1<⋯<푖푘≤푛. It is an exercise to check that this gives a valid structure. 푦 7. A non-example: define an equivalence relation (푥, 푦) ∼ (휆푥, 휆 ) for all 휆 ≠ 0 and define 푋 ∶= R2/ ∼. {푥푦 = 푐} is one equivalence class if 푐 ≠ 0. If 푐 = 0, {푥푦 = 0} is three classes {(푥, 0) ∶ 푥 ≠ 0}, {(0, 0)}, {(0, 푦) ∶ 푦 ≠ 0}. Thus 푋 ≅ (−∞, 0) ∪ {0′, 0″, 0‴} ∪ (0, ∞). Define charts (푖) 휑푖 ∶ (−∞, 0) ∪ 0 ∪ (0, ∞) → R in the obvious ways. We can check that it gives 푋 a valid 퐶∞ structure, except that the induced topology is non-Hausdorff! 8. For a non-second countable example, see example sheet 1 Q12. Definition (smooth map). Let 푀, 푁 be manifolds, a continuous map 푓 ∶ 푀 → 푁 is smooth or 퐶∞ if for any 푝 ∈ 푀, there exists charts (푈, 휑) on 푀, (푉 , 휓) on 푁 such that 푝 ∈ 푈, 푓(푝) ∈ 푉 and 휓 ∘ 푓 ∘ 휑−1 is smooth wherever it is defined, i.e. on 휑(푈 ∩ 푓−1(푉 )) ⊆ R푛 where 푛 = dim 푀. Remark. Note that by continuity, 휑(푈 ∩푓−1(푉 )) is necessarily open. Of course we can do it differently by not requiring 푓 to be continuous a priori but instead ask the above set to be open. For any charts ̃휑, 휓,̃ 휓̃ ∘ 푓 ∘ ̃휑−1 = (휓̃ ∘ 휓−1) ∘ (휓 ∘ 푓 ∘ 휑−1) ∘ (휑 ∘ ̃휑−1) is a composition of smooth map so is smooth. Thus smoothness of a map is independent of charts. 4 1 Manifolds Definition (diffeomorphism). A smooth map 푓 ∶ 푀 → 푁 is a diffeomor- phism if 푓 is bijective and 푓−1 is smooth. If such 푓 exists, 푀 and 푁 are diffeomorphism. Remark. 1. Our definition of smoothness is a generalisation of that in calculus. More precisely, 푓 ∶ R푛 → R푚 is smooth if and only if it is smooth in the calculus sense. 2. Any chart 휑 ∶ 푈 → R푑 is a diffeomorphism onto its image. 3. Composition of smooth maps is smooth. 5 2 Matrix Lie groups 2 Matrix Lie groups 2 We can view GL(푛, R) as an array of numbers and thus embed it in R푛 . Fur- thermore, by considering the determinant function it is an open subset so is an 푛2-manifold. Matrix multiplication is obviously smooth. In the same vein we have GL(푛, C), SL(푛, R) etc as mannifolds with smooth multiplications. Definition (Lie group). A group 퐺 is a Lie group if 퐺 is a manifold and has a compatible group structure, i.e. the map (휎, 휏) ↦ 휎휏 −1 is smooth. Before Lie groups, let’s have a short digression in analysis to discuss the exponential map.
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