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Geometry and Group Theory Geometry and Group Theory ABSTRACT In this course, we develop the basic notions of Manifolds and Geometry, with applications in physics, and also we develop the basic notions of the theory of Lie Groups, and their applications in physics. Contents 1 Manifolds 3 1.1 Some set-theoretic concepts . .... 3 1.2 TopologicalSpaces ............................... 4 1.3 Manifolds ..................................... 5 1.4 Tangentvectors.................................. 13 1.5 Co-vectors..................................... 18 1.6 An interlude on vector spaces and tensor products . ........ 21 1.7 Tensors ...................................... 22 1.8 TheMetricTensor ................................ 25 1.9 Covariant differentiation . 28 1.10 TheRiemanncurvaturetensor . 36 1.11 DifferentialForms ................................ 46 1.12 Integration, and Stokes’ Theorem . ...... 49 1.13 The Levi-Civita Tensor and Hodge Dualisation . ........ 55 1.14 The δ OperatorandtheLaplacian . 62 1.15 Spin connection and curvature 2-forms . ....... 66 2 General Relativity; Einstein’s Theory of Gravitation 73 2.1 TheEquivalencePrinciple . 73 2.2 ANewtonianInterlude.. .. .. .. .. .. .. .. 77 2.3 TheGeodesicEquation ............................. 78 2.4 TheEinsteinFieldEquation. 82 2.5 TheSchwarzschildSolution . 86 2.6 OrbitsAroundaStarorBlackHole . 90 3 Lie Groups and Algebras 97 3.1 DefinitionofaGroup............................... 97 3.2 LieGroups .................................... 99 3.3 TheClassicalGroups.............................. 105 3.4 LieAlgebras.................................... 115 3.5 RootsandWeights ................................ 121 3.6 Root Systems for the Classical Algebras . 161 1 The material in this course is intended to be more or less self contained. However, here is a list of some books and other reference sources that may be helpful for some parts of the course: 1. J.G. Hocking and G.S. Young, Topology, (Addison-Wesley, 1961). This is a very mathematical book on topological spaces, point-set topology, and some more advanced topics in algebraic topology. (Not for the faint-hearted!) 2. T. Eguchi, P.B. Gilkey and A.J. Hanson. Gravitation, Gauge Theories and Differen- tial Geometry, Physics Reports, 66, 213 (1980). This is a very readable exposition of the basic ideas, aimed at physicists. Some portions of this course are based fairly ex- tensively on this article. It also has the merit that it is freely available for downloading from the web, as a PDF file. Go to http://www.slac.stanford.edu/spires/hep/, type ”find a gilkey and a hanson”, and follow the link to Science Direct for this article. Note that Science Direct is a subscription service, and you must be connecting from a URL in the tamu.edu domain, in order to get free access. 3. H. Georgi, Lie Algebras and Particle Physics, Perseus Books Group; 2nd edition (September 1, 1999). This is quite a useful introduction to some of the basics of Lie algebras and Lie groups, written by a physicist for physicists. It is a bit idiosyncratic in its coverage, but what it does cover is explained reasonably well. 4. R. Gilmore, Lie Groups Lie Algebras and Some of Their Applications, John Wiley & Sons, Inc (1974). A more complete treatment of the subject. Quite helpful, especially as a reference work. 2 1 Manifolds One of the most fundamental constructs in geometry is the notion of a Manifold. A manifold is, in colloquial language, the arena where things happen. Familiar examples are the three- dimensional space that we inhabit and experience in everyday life; the surface of a ball, viewed as a two-dimensional closed surface on which, for example, an ant may walk; and the four-dimensional Minkowski spacetime that is the arena where special relativity may be formulated. In order to give a reasonably precise description of a manifold, it is helpful first to give a few rather formal definitions. It is not the intention in this course to make everything too formal and rigorous, so we shall try to strike a balance between formality and practical utility as we proceed. In particular, if things seem to be getting too abstract and rigourous at any stage, there is no need to panic, because it will probably just be a brief interlude before returning to a more intuitive and informal discussion. In this spirit, let us begin with some formal definitions. 1.1 Some set-theoretic concepts A set is a collection of objects, or elements; typically, for us, these objects will be points in a manifold. A set U is a subset of a set V if every element of U is also an element of V . If there exist elements in V that are not in the subset U, then U is called a proper subset of V . If U is a subset of V then the complement of U in V , denoted by V U, is the set of − all elements of V that are not in U. If U is a subset but not a proper subset, then V U − contains no elements at all. This set containing no elements is called the empty set, and is denoted by . By definition, therefore, is a subset of every set. ∅ ∅ The notion of the complement can be extended to define the difference of sets V and U, even when U is not a subset of V . Thus we have V U = x : x V and x / U . (1.1) \ { ∈ ∈ } If U is a subset of V then this reduces to the complement defined previously. Two sets U and V are equal, U = V , if every element of V is an element of U, and vice versa. This is equivalent to the statement that U is a subset of V and V is a subset of U. From two sets U and V we can form the union, denoted by U V , which is the set of ∪ all elements that are in U or in V . The intersection, denoted by U V , is the set of all ∩ elements that are in U and in V . The two sets U and V are said to be disjoint if U V = , ∩ ∅ i.e. they have no elements in common. 3 Some straightforwardly-established properties are: A B = B A , A B = B A , ∪ ∪ ∩ ∩ A (B C) = (A B) C , A (B C) = (A B) C , (1.2) ∪ ∪ ∪ ∪ ∩ ∩ ∩ ∩ A (B C) = (A B) (A C) , A (B C) = (A B) (A C) . ∪ ∩ ∪ ∩ ∪ ∩ ∪ ∩ ∪ ∩ If A and B are subsets of C, then C (C A)= A, C (C B)= B , − − − − A (A B)= A B , − \ ∩ C (A B) = (C A) (C B) , − ∪ − ∩ − C (A B) = (C A) (C B) . (1.3) − ∩ − ∪ − 1.2 Topological Spaces Before being able to define a manifold, we need to introduce the notion of a topological space. This can be defined as a point set S, with open subsets , for which the following Oi properties hold: 1. The union of any number of open subsets is an open set. 2. The intersection of a finite number of open subsets is an open set. 3. Both S itself, and the empty set , are open. ∅ It will be observed that the notion of an open set is rather important here. Essentially, a set X is open if every point x inside X has points round it that are also in X. In other words, every point in an open set has the property that you can wiggle it around a little and it is still inside X. Consider, for example, the set of all real numbers r in the interval 0 <r< 1. This is called an open interval, and is denoted by (0, 1). As its name implies, the open interval defines an open set. Indeed, we can see that for any real number r satisfying 0 <r< 1, we can always find real numbers bigger than r, and smaller than r that still themselves lie in the open interval (0, 1). By contrast, the interval 0 < r 1 is not open; ≤ the point r = 1 lies inside the set, but if it is wiggled to the right by any amount, no matter how tiny, it takes us to a point with r> 1, which is not inside the set. Given the collection of open subsets of S, we can define the notion of the limit {Oi} point of a subset, as follows. A point p is a limit point of a subset X of S provided that 4 every open set containing p also contains a point in X that is distinct from p. This definition yields a topology for S, and with this topology, S is called a Topological Space. Some further concepts need to be introduced. First, we define a basis for the topology of S as some subset of all possible open sets in S, such that by taking intersections and unions of the members of the subset, we can generate all possible open subsets in S. An open cover U of S is a collection of open sets such that every point p in S is contained in { i} at least one of the Ui. The topological space S is said to be compact if every open covering U has a finite sub-collection U , ,U that also covers S. { i} { i1 · · · in } Finally, we may define the notion of a Hausdorff Space. The topological space S is said to obey the Hausdorff axiom, and hence to be an Hausdorff Space, if, for any pair of distinct points p and p in S, there exist disjoints open sets amd , each containing just one 1 2 O1 O2 of the two points. In other words, for any distinct pair of points p1 and p2, we can find a small open set around each point such that the two open sets do not overlap.1 We are now in a position to move on to the definition of a manifold. 1.3 Manifolds Before giving a formal definition of a manifold, it is useful to introduce what we will recognise shortly as some very simple basic examples.
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