5 3 2 Self duality 23,5 3 3 Action principles 23,5 3 4 The Poincare Lemma 24. 6 Maps between manifolds 24,6 1 The push forward map 25. 6 2 The pull back map 25, 6 3 Exterior derivative commutes with pull back 25. 6 4 Diffeomorphisms Active versus passive viewpoint 26. 6 5 Invariant tensor fields 26,6 6 Immersions and embeddings 27. 7 One parameter families of diffeomorphims,Lie derivatives 27. 7 0 1 Example Functions 28,7 0 2 Example Vector fields 28. 7 0 3 Lie derivative of covariant tensors 29, 7 0 4 Example Adapted coordinates stationary and static metrics 29. 7 1 Lie derivative and Lie bracket 30,7 1 1 The Jacobi identity 30. 7 1 2 Example diff S 1 is called the Virasoro algebra 30. 7 2 The Lie bracket and closure of infinitesimal rectangles 30. 7 3 Frobenius Theorem 31, 7 3 1 Example The simplest non trivial Lie algebra 31. 7 3 2 Example Two commuting translations 31,7 3 3 Example Kaluza or torus reductions 31. 7 3 4 Example so 3 32,7 4 Lie derivative and exterior derivative 32. 7 5 Cartan s formula Lie derivative and interior product 32. 7 5 1 Example Electrostatic potentials 32,8 Affine connections 33. 8 1 Introduction of a basis Moving Frames 33, 8 2 Covariant Derivative of an arbitrary Representation GL n R 34. 8 3 Effect of a change of basis 34,8 4 The Torsion Tensor of an Affine connection 35. 8 5 Geometrical Interpretation of Torsion 36, 8 5 1 Example The right and left connections on a Lie Group 36. 8 6 The Levi Civita connection 36, 8 6 1 Example metric preserving connections with torsion 37. 8 6 2 Example Weyl Connections 37,8 7 Holonomy and Autoparallels 38. 8 7 1 Holonomy Group of an Affine Connection 39,8 8 Autoparallel curves 39. 8 8 1 Example Auto parallels on Lie Groups 40, 8 9 Examples of connections with torsion Tele parallelism 40. 8 10 Projective Equivalence 40, 8 10 1 Example metric preserving connections having the same geodesics 41. 9 The Curvature Tensor 41, 9 1 Flat coordinates for flat torsion free connections 42. 9 1 1 Example Beltrami coordinates for S n and De Sitter spacetimes 42. 9 1 2 Bianchi Identities 43,9 1 3 Ricci Tensor 43,9 1 4 The other contraction 43. 9 1 5 Example Weyl connections 43,10 Integration on manifolds Stokes Theorem 43. 10 0 6 The divergence operator 44,10 1 The Brouwer degree 45. 10 1 1 The Gauss linking number 45,10 1 2 The Gauss Bonnet Theorem 45. 10 2 A general framework for classical field and brane theories 46. 10 2 1 Particles 46,10 2 2 Strings membranes and p branes 46. 10 2 3 Non linear sigma models 47,10 2 4 Topological Conservation laws 47. 10 2 5 Coupling of p branes to p 2 forms 48,10 2 6 Topological defects 49. 10 2 7 Self interactions of p forms 49,11 Actions of groups on manifolds 50. 11 1 Groups and semi groups 50, 11 2 Transformation groups Left and Right Actions 50. 11 3 Effectiveness and Transitivity 51,11 4 Actions of groups on themselves 51. 11 5 Cosets 51,11 6 Orbits and Stabilizers 52, 11 6 1 Example De Sitter and Anti de Sitter Spacetimes 52. 11 6 2 Example Complex Projective space 52,11 7 Representations 53. 11 7 1 Reducible and irreducible representations 53. 11 7 2 The contragredient representation 53,11 7 3 Equivalent Representations 54. 11 7 4 Representations on functions 54, 11 8 Semi direct products group extensions and exact sequences 55. 11 8 1 The five Lorentz groups 56,11 9 Geometry of Lie Groups 56. 11 9 1 Matrix Groups 57, 11 10Infinitesimal Generators of Right and Left Translations 57. 11 10 1 Example Matrix groups 58, 11 11Right invariant Vector Fields generate Left Translations and vice versa 58. 11 12Lie Algebra 58,11 13Origin of the minus sign 58. 11 14Brackets of left invariant vector fields 59, 11 14 1 Example Right and Left vector fields commute 59. 11 15Maurer Cartan Forms 59,11 15 1 Example Matrix groups 59. 11 16Metrics on Lie Groups 60,11 16 1 Example SU 2 61. 11 17Non degenerate Killing forms 61, 11 17 1 Necessary and sufficient condition that the Killing form is non degenerate 61. 11 17 2 Invariance of the Killing metric under the co adjoint action 61. 11 17 3 Signature of the Killing metric 62,11 17 4 Example SO 3 and SU 2 62. 11 17 5 Matrix Groups 62,11 17 6 Example SL 2 R 62. 11 17 7 Example SO 2 1 SL 2 R Z2 62,11 17 8 Example Nil or the Heisenberg Group 63. 11 17 9 Example Two dimensional Euclidean Group E 2 64. 11 18Rigid bodies as geodesic motion with respect to a left invariant metric on SO 3 64. 11 18 1 Euler angles 65,11 18 2 SL 2 R and the Goedel Universe 67. 11 18 3 Example Minkowski Spacetime and Hermitian Matrices 67. 11 18 4 Example SO 4 and quaternions 68, 11 18 5 Example Hopf fibration and Toroidal Coordinates 68. 11 18 6 Example Kaluza Klein Theory 69,12 Fibre Bundles 70. 12 0 7 Example Frame bundle and associated Tangent bundle 71. 12 0 8 Group extensions as Principal bundles 72,12 1 Global right action on a Principal bundle 72. 12 1 1 Examples Coset spaces and monopoles 73,12 2 Reductions of Bundles 73. 12 2 1 Reduction of the frame bundle 73,12 2 2 Existence of Lorentzian metrics 73. 12 2 3 Time orientation 74,12 3 Global and Local Sections of Bundle 74. 12 3 1 Globally Hyperbolic four dimensional spacetimes are pararlellizable 75. 12 3 2 Euler number of a vector bundle 75,12 3 3 Global sections of principal bundles 75. 12 4 Connections on Vector Bundles 75,12 4 1 Form Notation 76. 12 4 2 Metric preserving connections 76,13 Symplectic Geometry 76. 13 0 3 Poisson Brackets 78,13 0 4 A generalization Poisson Manifolds 78. 13 0 5 Poisson manifolds which are not Symplectic 79. 13 0 6 Co adjoint orbits 79,13 0 7 The canonical one form 80. 13 0 8 Darboux s Theorem and quantization 80,13 0 9 Lie Bracket and Poisson Bracket 80. 13 0 10 Canonical transformations as symplecto morphisms 81. 13 0 11 Infinitesimal canonical transformations Poincare s Integral Invariants 81. 13 1 Hamiltonian Symmetries and moment maps 82,13 1 1 Geodesics and Killing tensors 83. 13 1 2 Lie Bracket and Poisson algebra s of Killing fields 84. 13 1 3 Left invariant metrics and Euler equations 84. 13 1 4 Liouville Integrable Systems 85,13 2 Marsden Weinstein Reduction 85. 13 2 1 Example CPn and the isotropic harmonic oscillator 86. 13 2 2 Example The relativistic particle 86,13 2 3 Reduction by G 86. 13 3 Geometric Quantization 87,1 Contents of the Course. I Differential Forms, I 1 Grassmann Algebra Interior Product and Wedge Product. I 2 Hodge Duality,1 3 Exterior Differentiation,1 4 Behaviour under pull back. 1 5 Stokes s Theorem,II Action of Groups on Manifolds. II 1 Definition and Elementary Properties of Group Actions. II 2 Homogeneous Spaces and Co set spaces,II 3 Left and Right Actions on Groups. II 4 Representations of groups,III Geometry of Lie Groups. III 1 Left and right invariant vector fields,III 2 Exponential Map. III 3 Cartan Maurer Equations,III 4 Connections and Metrics on Lie Groups. III 5 Geodesics and Auto parallels on Lie Groups,IV Fibre Bundles. IV 1 Definition of Fibre bundles, IV 2 Principal Bundles Vector Bundles Associated Bundles. IV 3 Connections on Bundles Curvature and Cartan s Equations. V Symplectic Geometry and Mechanics,V 1 Hamiltonian Mechanics. V 2 Poisson and Symplectic Manifolds, V 3 Hamiltonian Symmetries Poisson Brackets and Lie Brackets. V 4 Moment maps and Hamiltonian Reduction,V 6 Elementary ideas about Geometric Quantization. These foregoing notes consist of an introductiion to differential geometry at the level needed to un. derstand the course followed by more material on the content of the lectures in roughly the order it. will be lectured The introductory material will almost all have been covered in the Michaelmas term. Part III General Relativity Course Similar material although in a somewhat different style is covered. in the Michaelmas term pure mathematics course on Part III Differential Geometry For an elementary. account of General Relativity in old fashioned Tensor Calculus notation the reader may consult my Part. II lecture notes which are available on my Web site. The lecture notes are the result of combining and extending previous lecture notes and are not in a. final polished form As a result various topics are treated in a less than optimal order and not all mistakes. and typo s have yet been eliminated Thus they are not designed as a subsitute for taking notes during. the lectures merely as a reference In fact a transcript of a previous year s lectures was prepared by one. of the audience using microsoft word This is available on my web page. http www damtp cam ac uk user gr members gwg html,to users inside DAMTP. I shall be available in my office B1 24 from 2 00pm to 3 00pm on most Tuesdays to answer queries. A sign up sheet for supervisions will be circulated during lecture 3. 2 Some Textbooks,In preparing the course I have used among others. 1 Y Choquet Bruhat C DeWitt Morette M Dilliard Bleick Analysis Manifolds and Physics. North Holland,2 H Flanders Differential Forms Dover. 3 B O Neil Semi Riemannian Geometry Academic Press. 4 B Dubrovin S Novikov A Fomenko Modern Geometry Springer. 5 T Eguchi P Gilkey and A J Hanson Physics Reports 66 1980 213 393. 6 V Arnold Mathematical Methods of Classical Mechanics Springer. 7 N M J Woodhouse Geometric Quantization Oxford, This list does not exhaust the set of good textbooks on this subject at the level at which it will be. lectured In particular many speak highly of, 8 C Nash and S Sen Topology and Geometry for physicists Academic Press. 9 M Nakahara Geometry Topology and Physics Institute of Physics. 10 R Darling Differential Forms and Connections Cambridge University Press. 11 T Fraenkel The Geometry of Physics Cambridge University Pres. There is a great deal of relevant material on particular topics in. 12 J Marsden and R Ratiu Introduction to Mechanics and Symmetry. 13 R Gilmore Lie Algebra Lie Groups and some of their Applications. 14 R Aldrovandi and J G Perira An Introduction to Geometrical Physics World Scientific. 15 B Felsager Geometry Particles and Physics, 16 C J Isham Modern Differential Geometry for Physicists World Scientific. 17 V Guillemin and S Sternberg Symplectic Techniques in Physics Cambridge University Press. 18 R Abraham and J Marsden Foundations of Mechanics. 19 A S Schwarz Topology for Physicists Springer, 20 A Visconti Introductory Differential Geometry for Physicists. Perhaps the best book covering almost all the course is probably number 1 or number 4 I found 2. extremely useful for a first look at differential forms Finally much information and many relevant. examples are contained in, 21 C Misner K Thorne and J Wheeler Gravitation Freeman. 3 Manifolds, The reader is expected to have encountered already some of the more elementary ideas of Einstein s theory. of General Relativity and with it the basic concepts of tensor analysis For advanced work however. rather more is required and one often needs to consider general topologically non trivial manifolds that. may not be directly related to spacetime the target spaces of non linear sigma models for example What. follows is a brief introduction to the necessary machinery The underlying theme is to start with a bare. manifold with no further structure and then to introduce such things as connections and metrics. It is customary to utter the following incantation. Spacetime is a connected Hausdorff differentiable pseudo Riemannian manifold of dimension 4 whose. points are called events, What does this mean We need to introduce some ideas. 3 1 Topological spaces, A topological space X consist of a point set X where Ui i I where I is an index set labelling. members of a a collection of special subsets of X called open sets such that. i All unions of open sets are open,ii finite intersections of open sets are open and. iii X and the empty set are open, Different collections of subsets may endow the same point set X with a different topology A basis. for a topology is a subset of all possible open sets which by intersections and unions can generate all. possible open sets A possible basis for the real line R is the collection of open intervals A possible. basis for Rn is the collection of open balls indexed by centre x0 and radius r such that x x0 r An. open cover Ui of X is a collection of open sets such that every point x X is contained in at least. one Ui X is compact if every open cover has a finite sub cover X is Hausdorff if every pair of disjoint. points is contained in a disjoint pair of open sets Any open set containing a point x X is also called a. neighbourhood of x, A function from one topological space X to another Y f X Y is continous if the inverse image of. every open set is open Two topological spaces are homeomorphic if there is 1 1 map from X onto Y a. bijection such that and 1 are continuous By Leibniz s principle of the Identity of Indiscernibles. two homeomorphic topological spaces are usually thought to be same. 3 2 Manifolds, We define a smooth1 n manifold with a smooth atlas of charts as. i a topological space X,ii an open cover Ui called patches and. iii a set called an atlas of maps i Ui Rn called charts which are injective homeomorphisms. onto their images and whose images are open in Rn, iv if two patches U and U intersect then on U U both 1 and 1 are smooth maps from. We write x x 1 2 n x is called a local coordinate on X In the usual notation we have. 1 given by x x and 1 given by x x, Note that we are here taking the passive viewpoint of coordinate transformations thinking of the. points of the manifold as fixed and only their labels as changing. Two atlases are said to be compatible if where defined the coordinates are smooth functions of one. 1 Smooth means as smooth as we need When in doubt take it to mean C. A smooth n manifold with complete atlas is the maximal equivalence class consisting of all possible. compatible atlases, We shall denote such a manifold by M or M n As we have defined it a manifold need not be Hausdorff. and so this as an additional requirement,3 2 1 Example S 2 and stereographic projection. Consider the unit two sphere which we may think as the surface. x2 y 2 z 2 1 3 1, embedded R3 We could use as coordinates spherical polars However these break down at the. north and south poles where 0 and respectively because one cannot assign to these points. unique values of However we can use stereographic projection to define two charts. U S 2 south pole 7 x iy ei tan 3 2,U S 2 north pole 7 x iy e i cot 3 3. On the overlap, This is a two chart atlas We get a six chart atlas by considering orthogonal projection with respect to. all three axes We think of S 2 as x2 y 2 z 2 1 and define. 3 x y z z 0 7 x y 3 5,3 x y z z 0 7 x y 3 6,3 2 2 Example Two incompatible atlases for R. Let z be the standard coordinate on R We get a one chart atlas by taking as to be the identity map. z 7 x say and another one chart atlas by taking for the cubic map z 7 x 3 Now x x1 3 is. not smooth at x 0,3 2 3 Example A non Hausdorff 1 manifold. Let X R 0 R 1 with x 0 and x 1 identified if x 0 The points 0 0 and 0 1 are. distinct but every pair of neigbourhoods intersects This is a toy model for a trouser universe which. splits into two The only one dimensional Hausdorff manifolds are S 1 and R. 3 2 4 Functions on manifolds and orientability, We can now define a real valued smooth function f M R as one which is smooth in all coordinate. systems i e f 1 f x is smooth in the usual sense The set of all smooth functions C M. on a manifold forms a commutative ring since we can define addition and multiplication by pointwise. addition and multiplication of the values We can also think of the functions as a commutative algebra. and we shall then denote them by F M, A manifold M is said to orientable if it admits an atlas such that for all overlaps the Jacobian satisfies. 3 2 5 Example Projective spaces, Consider first the real projective plane RP2 This is the set of directions through the origin in R2 i e. triples x y z such that x y z x y z 6 0 We get a three chart atlas as follows. a z a z if z 6 0,Set b y b y if y 6 0 3 8,c x a x if x 6 0. If zy 6 0 we have b1 1 a2 b2 a1 a2 Computing the Jacobian shows that this atlas is not oriented. In fact no oriented atlas exists RP2 is not orientable A similar calculation for RP3 does give an oriented. atlas wherefore RP3 is orientable The general results is that RPn is orientable if and only if n is odd. This may be understood by from the fact that RPn S n Z2 where Z2 acts by the antipodal map This. is orientation reversing if n is even and orientation preserving if n is odd. There is only one one dimensional compact Hausdorff manifold and hence the simplest projective. space RP1 is just S 1 We need two charts with coordinates x and x x1 RP1 is a homogeneous space. of P SL 2 R which acts by fractional linear transformations. x 7 ad bc 1 3 9, These transformations are also known as projective transformations and arise in string theory as a finite. dimensional subgroup of the infinite dimensional group of all diffeomorphisms of the circle S 1. 3 3 Tangent vectors, A smooth curve c in M is a smooth map c R M In local coordinates c 7 x where x is a. smooth function of A closed curve is a map from S 1 to M A curve is simple if it is 1 1 onto its image. For us a path is the image of a curve i e it is a point set Thus a curve contains information about. the parameterization that a path does not contain2 If we orient R we can give a privileged direction to. the associated equivalence class of curves and indicate this by attaching an arrow to the curve In other. words two curves x and x have the same orientation if say is a monotonically increasing. function of d d 0 If M is a spacetime a curve its path is called a world line and corresponds to. a particle The curve with opposite orientation x corresponds to its antiparticle 3. Given a curve c and a function f we can compose them to get a map f c R R given in local. coordinates by f x and we can differentiate it By the chain rule. If we look at this at a point p M and vary the curves passing through that point we get a map. T C M R taking smooth functions on M to the reals,T f 7 T f where x 0 p 3 11. which satisfies,i T f g T f T g linearity and,ii T f g T f g f T g the Leibniz rule. Such a map is called a tangent vector at p For clarity we could have written Tp but have not done in. order to eliminate clutter Since T is a linear operator we can add two such vectors according to the. T1 T2 f T1 f T2 f 3 12, 2 This terminology is not quite standard Many books use the word path for what we call a curve and appear to have no. special term for what we call a path, 3 This is independent of whether the spacetime itself has a time orientation. and we can multiply by constants Thus the space of tangent vectors at a point p M is a vector space. which we call the tangent space and denote by Tp M 4 Its dimension is n as may be seen by by using. Taylor s Theorem in local coordinates,f x f p x p f 3 13. T x T then 3 14,Tf T f p 3 15,and thus x is a basis and we may write. The basis x is called a coordinate basis or sometimes natural basis If T is the tangent vector to a curve. c we then have from 3 10, In a new coordinate system the components will change If. T T T 3 18,then chain rule gives, This is the elementary definition of a contravariant vector. We can now introduce the idea of a vector field which is a continuous assignment of a vector V p. Tp M to each point p in the manifold M In local coordinates. V V x 3 20, The set of all vector fields on M is denoted by T M or X M. Given a vector field V X M we have at least locally the associated integral curves defined as the. solutions of the non linear o d e, whose tangent vectors coincide with the vector field at every point in M In general a family of curves. one passing through each point p of M is called a congruence of curves. We can also introduce the concept of the tangent bundle T M or velocity space as the space of all. possible vectors at all possible points i e,T M Tp M 3 22. This is an 2n dimensional manifold with local coordinates x V where V V x One may think. of a vector field as a sort of n dimensional surface in T M In the terminology of vector bundles which. we will introduce in more detail later this surface is called a section. 3 4 Non coordinate bases, It is often useful to use for Tp M a non coordinate basis ea a 1 2 n called variously a tetrad. vierbein vielbein 4 leg or repe re mobilie moving frame etc Thus. T T a ea 3 23,4 Many authors omit the brackets and write Tp M. With respect to a coordinate basis x we have,ea ea x 3 24. It is customary to refer to as a world index and to a as a tangent space index This is because now we. are allowed not only coordinate transformations which induce a change of the basis x but also position. dependent changes of the basis ea,ea e a ab x eb 3 25. This induces the change of components,T a T a ab x T b where 3 26. ab cb ca 3 27, In a matrix notation in which the components T a form a column vector also denoted by T and the basis. vectors ea a row vector denoted by e,ab ab 3 28,cb 1 bc 3 29. T T e e 1 GL n R 3 30, A coordinate transformation induces a change of the natural basis with. However this is not the most general local frame rotation since. 3 5 Co vectors, Given any finite dimensional vector space V we define its dual space V as the space of linear maps. V R We write for V and for U W V,U h U i h U i R 3 34. h U W i h U i h W i and 3 35,h U i h U i R 3 36, This is a vector space of the same dimension as V and the dual of V is the original vector space V. Given a basis ea for V we set a ea and we denote the dual or reciprocal basis by ea such. hea eb i ba 3 37,U a U a 3 38,where U a are the components of U in the basis ea. 3 5 1 Examples of dual vector spaces, In homely three dimensions the dual space corresponds to momentum or wave vector space x k. is a scalar Given a basis e1 e2 e3 not necessarily orthonormal the reciprocal basis is given by. e3 e1e e 2 e3, etc This construction is frequently used in crystallography when e1 e2 e3 are the basis. vectors of a lattice The reciprocal basis gives the reciprocal lattice in momentum space. 3 6 Change of basis,Under a change of basis,ea ab eb 3 39. a ac c c 1 ca 3 40, Geometrically while vectors define directions through the origin of V one may think of co vectors as. hyperplanes or or co normals to planes through the origin This is because for fixed the set of points. U V such that X 0 is a hyperplane and and define the same hyperplane The duality is. then the standard one of points and hyperplanes in elementary projective geometry. Now at every point p in a manifold M one defines the cotangent space Tp M as the dual space of. the tangent space Tp M Then a covector field is one for which. f U f U 3 41,where now f is a function of x, An example of a co vector is a differential 1 form We consider a function f M R and define its. differential exterior derivative or gradient df by. hdf U i U f for all U X M 3 42, If we call 1 M the space of 1 forms on a manifold and 0 M the real valued functions on M. i e what thought of as a commutative ring we called C M and F M earlier then we have a map. d C M 0 M 1 M that is Leibnizian,d f g df g gdf f g 0 M 3 43. In a coordinate basis,df f x dx U x 3 44,hdf U i U f U x f x 3 45. df U U f 3 46, By arguments similar to those given for vector fields we deduce that dx gives a basis for Tp M dual. to the coordinate basis of vectors,Thus if dx under a coordinate transformation. which corresponds to the elementary definition of a covariant vector Of course not all 1 forms are of. the form df A necessary condition is that, We shall have more to say about sufficient conditions when we discuss the Poincare Lemma. The geometrical interpretation of df is as follows Locally at least f x const a defines the a. level sets of the function i e the n 1 dimensional surface a on which the function takes the constant. value a Now along a curve c intersecting a we have. T f hdf T i where 3 51, is the tangent to the curve c Now if c t lies in a d 0 and so the tangent vector T lies in. T and thus,hdf T i 0 3 53, Thus n x should be thought of as the co normal of the surface a Note that in the absence of. a natural map from co vectors to vectors a surface has no natural unique normal vector Such a map is. provided by a metric on M which we will introduce in a later section. Just as we defined the tangent bundle T M we can define the 2n dimensional co tangent bundle. T M as the space of a possible co vectors at all possible points. T M Tp M 3 54, This is also called the phase space or momentum space space of the manifold It has as local coordinates. x p where p are the components of an arbitrary 1 form p p dx. 3 7 Tensor algebra, Having defined vectors and co vectors we can go on to define the associated tensor product spaces Thus. a co tensor of rank q can be thought of as a multi linear map from the q fold Cartesian product. Tp M Tp M Tp M taking its values in R We write this as U V W or h U V W i. For co tensor fields the linearity is over functions i e. h f U gV hW i f gh h U V W i f g h C M 3 55,In a basis we have. ab c ea eb ec with 3 56,ab c ea eb ec 3 57, We can also use a coordinate basis in which the components of are. ab c ea eb ec 3 58, Similarly we can define contravariant and mixed tensors of type q. The usual operations of contraction symmetrization and anti symmetrization can be introduced. Special interest attaches to the latter since we can also introduce a product called the wedge product. Since the construction is both useful and universal and works for any vector space we devote the next. subsection to it,3 8 Exterior or Grassmann algebra. Given any n dimensional vector space V a p form is a totally antisymmetric multilinear map V V. V R We call p V the vector space of p forms and dim p V p n p By convention. V R We denote the direct sum of the vector spaces V by. V p V 3 59,Thus dim V 2n, We now turn V into an algebra over R by defining a product called the wedge or exterior product. p V q V p q V 3 60,which satisfies the following properties. i f f f R linearity,ii distributivity,iii associativity and. iv p q 1 pq q p graded commutativity, In rule iv p p V and q q V Note that V is what is called a graded vector space that is. it splits into a sum of vector spaces labelled by a degree p and wedge multiplication respects the grading. A coarser grading so called Z2 grading is obtained by lumping together even and odd degree forms. V V V Exterior multiplication also respects this grading. The explicit form of the wedge product is given by in components by. p q a1 ap ap 1 ap q a1 ap ap 1 ap q 3 61, where if eb and ea are mutually dual bases for V and V respectively such that hea eb i ba and. ea1 eap a1 ap 3 62,a1 ap ea1 ea2 eap a a ea1 ea2 eap 3 63. e1 e2 e1 e2 e2 e1 3 64,and for example the Faraday tensor is given by. F F dx dx Ei dt dxi ijk Bk dxi dxj 3 65,As an algebra V is generated by the relations. ea eb eb ea 0 3 66, In this context ea are often called Grassmann numbers and one calculates with them using the usual. rules of algebra taking into account the fact that they anti commute An arbitrary real valued function. of these Grassmann variables is a polynomial of at most degree n and is just another name for an element. of V The polynomials of degree p are just p forms i e elements of p V. 4 Inner products and pseudo Riemannian manifolds, In this section we shall introduce the idea of an inner product or metric on a manifold This is a. smooth assignment to the tangent space at each point of the manifold of an inner product or bilinear. form which is linear over functions To see what this means recall that an inner product or metric on. a finite dimensional vector space V is a real valued non degenerate symmetric bilinear form g i e a. function g V V R satisfying,i g U W g W U for all vectors U W V symmetry. ii g f1 U f2 W f1 f2 g U W for all U W V f1 f2 R linearity. iii g U W 0 W iff U 0 non degeneracy, In ii f1 f2 are arbitrary real numbers To gain insight we introduce a basis ea in which the metric. has components which we may think of as a symmetric matrix. gab g ea eb gba 4 67,Condition iii implies that,det gab 6 0 4 68. We may therefore diagonalize g and it will have s positive eigenvalues and t negative eigenvalues By. suitably rescaling we can find a pseudo orthonormal basis in which g is diagonal with entries 1 s times. and 1 t times One says that the metric has signature s t although sometimes s t is called the. signature Of course dim V s t, 4 1 The musical isomorphism Index raising and lowering. Given a bilinear form g which need be neither symmetric nor non degenerate we get a natural map. V V called index lowering since if U is a vector we can define a 1 form U by. hU W i g U W W V 4 69,In components,U b U a gab 4 70. If g is non degenerate this map is invertible and we get an isomorphism of V and V called the musical. isomorphism The inverse map V V is called index raising Thus if. g 1 ea eb g ab gab g bc bc,then given a 1 form we obtain a vector by. for all 1 forms 4 72, If a given metric is understood it is usual to drop the s and s and use the same kernel letter for. vectors or tensors related by index raising and lowering Indeed if g is positive it is customary to restrict. bases to be orthonormal in which,gab ab 4 74, and the distinction between contravariant and covariant tensors is dropped All indices are usually written. downstairs as in elementary Cartesian tensors and vectors in Euclidean 3 space E3 If g is indefinite one. typically but not always adopts a pseudo orthonormal basis for which. gab ab 4 75, with ab diag 1 1 1 1 1 1 However index position must still be retained since. factors of 1 arise on raising and lowering, In four dimensional general relativity it is sometimes convenient to adopt a null tetrad l n m m for. with m 12 a ib with the real dyad a b unit and orthogonal to one another and to the real null. vectors l and n,4 1 1 Moduli space of metrics, One may clearly endow a given vector space V with many metrics It suffices to give a dual basis ea. and deem it to be pseudo orthonormal Of course SO s t rotations of our basis that preserve ab will. give the same metric If we fix an initial basis any other basis may be specified by giving the element of. GL n R needed to pass to the new basis i e by the matrix with elements. The metric is thus,g ea ab eb g et e 4 78, Now e and Se with S SO s t give the same metric so the set of different metrics is the coset space. SO s t GL n R, In fact the space of metrics itself carries a one parameter family of GL n R metrics given by. ds2 Tr e 1 dee 1 de Tr e 1 de, Metrics on spaces of metrics of this kind arise frequently in general relativity particularly in connection. with dimensional reduction Another example arises in the Hamiltonian or 3 1 formulation of general. relativity in which the kinetic term may be considered a metric on the space of 3 metrics called in this.

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