This thesis is mainly focused on the equivalence problem for a subclass of Lorentzian manifolds: the degenerate Kundt spacetimes. These spacetimes are not defined uniquely by their scalar curvature invariants. To prove two metrics are diffeomorphic, one must apply Cartan's equivalence algorithm, which is a non-trivial task: in four dimensions Karlhede has adapted the algorithm to the formalism of General Relativity and significant effort has been spent applying this algorithm to particular subcases. No work has been done on the higher dimensional case. First, we study the existence of a non-spacelike symmetry in two well-known subclasses of the N dimensional degenerate Kundt spacetimes: those spacetimes with constant scalar curvature invariants (CSI) and those admitting a covariant constant null vector (CCNV). We classify the CSI and CCNV spacetimes in terms of the form of the Killing vector giving constraints for the metric functions in each case. For the rest of the thesis we fix N=4 and study a subclass of the CSI spacetimes: the CSI-? spacetimes, in which all scalar curvature invariants vanish except those constructed from the cosmological constant. We produce an invariant characterization of all CSI-? spacetimes. The Petrov type N solutions have been classified using two scalar invariants. However, this classification is incomplete: given two plane-fronted gravitational waves in which both pairs of invariants are similar, one cannot prove the two metrics are equivalent. Even in this relatively simple subclass, the Karlhede algorithm is non-trivial to implement. We apply the Karlhede algorithm to the collection of vacuum Type N VSI (CSI-?, ? = 0) spacetimes consisting of the vacuum PP-wave and vacuum Kundt wave spacetimes. We show that the upper-bound needed to classify any Type N vacuum VSI metric is four. In the case of the vacuum PP-waves we have proven that the upper-bound is sharp, while in the case of the Kundt waves we have lowered the upper-bound from five to four. We also produce a suite of invariants that characterize each set of non-equivalent metrics in this collection. As an application we show how these invariants may be related to the physical interpretation of the vacuum plane wave spacetimes.

Neste trabalho foi feito um estudo do limite de Karlhede para ondas pp. Para este fim, uma revisão rigorosa de Geometria Diferencial foi apresentada numa abordagem independente de sistemas de coordenadas. Além da abordagem usual, a curvatura de uma variedade riemanniana foi reescrita usando os formalismos de referenciais, formas diferenciais e espinores do grupo de Lorentz. O problema de equivalência para geometrias riemannianas foi formulado e as peculiaridades de sua aplicação é a Relatividade Geral são delineadas. O limite teórico de Karlhede para espaços-tempo de vácuo de tipo Petrov N foi apresentado. Esse limite é estudado na prática usando técnicas espinores e as condições para sua existência são resolvidas sem a introdução de sistemas de coordenadas. / In this work a study of the Karlhede limit was made. To this end, a thorough review of Differential Geometry was presented in a coordinate independent approach. Besides the usual approach, the curvature of a riemannian manifold was rewritten using the formalisms of frames, differential forms and Lorentz group spinors. The equivalence problem for riemannian geometries was formulated and the peculiarities of its application to General Relativity are outlined. The theoretical Karlhede limit for vacuum Petrov N space-times is presented. This limit was studied in practice using spinor techniques and the conditions for its existence are solved without introducing coordinate systems.

Neste trabalho foi feito um estudo do limite de Karlhede para ondas pp. Para este fim, uma revisão rigorosa de Geometria Diferencial foi apresentada numa abordagem independente de sistemas de coordenadas. Além da abordagem usual, a curvatura de uma variedade riemanniana foi reescrita usando os formalismos de referenciais, formas diferenciais e espinores do grupo de Lorentz. O problema de equivalência para geometrias riemannianas foi formulado e as peculiaridades de sua aplicação é a Relatividade Geral são delineadas. O limite teórico de Karlhede para espaços-tempo de vácuo de tipo Petrov N foi apresentado. Esse limite é estudado na prática usando técnicas espinores e as condições para sua existência são resolvidas sem a introdução de sistemas de coordenadas. / In this work a study of the Karlhede limit was made. To this end, a thorough review of Differential Geometry was presented in a coordinate independent approach. Besides the usual approach, the curvature of a riemannian manifold was rewritten using the formalisms of frames, differential forms and Lorentz group spinors. The equivalence problem for riemannian geometries was formulated and the peculiarities of its application to General Relativity are outlined. The theoretical Karlhede limit for vacuum Petrov N space-times is presented. This limit was studied in practice using spinor techniques and the conditions for its existence are solved without introducing coordinate systems.

Ce mémoire est une contribution à la résolution du problème d'équivalence pour les variétés de Cauchy-Riemann en dimension inférieure ou égale à 5. On traite d'abord du cas des variétés CR de dimension 5, qui sont 2-nondégénérées et de rang de Levi constant égal à 1. Pour une telle variété, on obtient deux invariants, J et W, dont l'annulation simultanée caractérise l'équivalence locale à une variété modèle, le tube au-dessus du cône de lumière. Si l'un des deux invariants ne s'annule pas, on construit un parallélisme absolu, i.e. on montre que le problème d'équivalence se réduit à un problème d'équivalence entre {e}-structures de dimension 5. On étudie ensuite le problème d'équivalence pour certaines variétés CR de dimension 4 appelées variétés de Engel. Ce problème est résolu par la construction d'une connexion de Cartan sur un fibré principal de dimension 5. On traite ensuite du cas de variétés CR de dimension 5 dont le fibré CR vérifie une certaine hypothèse de dégénérecence. Le problème d'équivalence est résolu dans ce cas par la construction d'une connexion de Cartan sur un fibré de dimension 6. Enfin, on détermine les algèbres de Lie des automorphismes infinitésimaux des modèles pour les trois classes de variétés CR étudiées. / This memoir contributes to solve the equivalence problem for CR-manifolds in dimension up to 5. We first deal with the equivalence problem for 5-dimensional CR-manifolds which are 2-nondegenerate and of constant Levi rank 1. For such a manifold M, we find two invariants, J and W, the annulation of which gives a necessary and sufficient condition for M to be locally CR-equivalent to a model hypersurface, the tube over the light cone. If one of the invariants does not vanish on M, we construct an absolute parallelism on M, that is we show that the equivalence problem reduces to an equivalence problem between 5-dimensional {e}-structures. We then study the equivalence problem for 4-dimensional CR-manifolds which are known as Engel manifolds. This problem is solved by the construction of a canonical Cartan connection on a 5-dimensional bundle through Cartan's equivalence method. We also study the equivalence problem for 5-dimensional CR-manifolds whose CR-bundle satisfies a certain degeneracy assumption, and show that in this case, the problem is solved by the construction of a Cartan connection on a 6-dimensional bundle. The last part of this memoir is devoted to the determination of the Lie algebra of infinitesimal automorphisms for the model manifolds of the three previous classes.

The analysis of similarity of matrices over fields, as well as integral domains which are not fields, is a classical problem in Linear Algebra and has received considerable attention. A related problem is that of simultaneous similarity of matrices. Many interesting algebraic questions that arise in such problems are discussed by Shmuel Friedland[1]. A special case of this problem is that of Simultaneous Unitary Similarity of hermitian matrices, which we describe as follows: Given a collection of m ordered pairs of similar n×n hermitian matrices denoted by {(Hl,Dl)}ml=1, 1. determine if there exists a unitary matrix U such that UHl U∗ = Dl for all l, 2. and in the case where a U exists, find such a U, (where U∗is the transpose conjugate of U ).The problem is easy for m =1. The problem is challenging for m > 1.The problem stated above is the algorithmic version of the problem of classifying hermitian matrices upto unitary similarity. Any problem involving classification of matrices up to similarity is considered to be “wild”[2]. The difficulty in solving the problem of classifying matrices up to unitary similarity is a indicator of, the toughness of problems involving matrices in unitary spaces [3](pg, 44-46 ).Suppose in the statement of the problem we replace the collection {(Hl,Dl)}ml=1, by a collection of m ordered pairs of complex square matrices denoted by {(Al,Bl) ml=1, then we get the Simultaneous Unitary Similarity problem for square matrices. Suppose we consider k ordered pairs of complex rectangular m ×n matrices denoted by {(Yl,Zl)}kl=1, then the Simultaneous Unitary Equivalence problem for rectangular matrices is the problem of finding whether there exists a m×m unitary matrix U and a n×n unitary matrix V such that UYlV ∗= Zl for all l and in the case they exist find them. In this thesis we describe algorithms to solve these problems. The Simultaneous Unitary Similarity problem for square matrices is challenging for even a single pair (m = 1) if the matrices involved i,e A1,B1 are not normal. In an expository article, Shapiro[4]describes the methods available to solve this problem by arriving at a canonical form. That is A1 or B1 is used to arrive at a canonical form and the matrices are unitarily similar if and only if the other matrix also leads to the same canonical form. In this thesis, in the second chapter we propose an iterative algorithm to solve the Simultaneous Unitary Similarity problem for hermitian matrices. In each iteration we either get a step closer to “the simple case” or end up solving the problem. The simple case which we describe in detail in the first chapter corresponds to finding whether there exists a diagonal unitary matrix U such that UHlU∗= Dl for all l. Solving this case involves defining “paths” made up of non-zero entries of Hl (or Dl). We use these paths to define an equivalence relation that partitions L = {1,…n}. Using these paths we associate scalars with each Hl(i,j) and Dl(i,j)denoted by pr(Hl(i,j)) and pr(Dl(i,j)) (pr is used to indicate that these scalars are obtained by considering products of non-zero elements along the paths from i,j to their class representative). Suppose i (I Є L)belongs to the class[d(i)](d(i) Є L) we denote by uisol a modulus one scalar expressed in terms of ud(i) using the path from i to d( i). The free variable ud(i) can be chosen to be any modulus one scalar. Let U sol be a diagonal unitary matrix given by U sol = diag(u1 sol , u2 sol , unsol ). We show that a diagonal U such that U HlU∗ = Dl exists if and only if pr(Hl(i, j)) = pr(Dl(i, j))for all l, i, j and UsolHlUsol∗= Dl. Solving the simple case sets the trend for solving the general case. In the general case in an iteration we are looking for a unitary U such that U = blk −diag(U1,…, Ur) where each Ui is a pi ×p (i, j Є L = {1,… , r}) unitary matrix such that U HlU ∗= Dl. Our aim in each iteration is to get at least a step closer to the simple case. Based on pi we partition the rows and columns of Hl and Dl to obtain pi ×pj sub-matrices denoted by Flij in Hl and Glij in D1. The aim is to diagonalize either Flij∗Flij Flij∗ and a get a step closer to the simple case. If square sub-matrices are multiples of unitary and rectangular sub-matrices are zeros we say that the collection is in Non-reductive-form and in this case we cannot get a step closer to the simple case. In Non- reductive-form just as in the simple case we define a relation on L using paths made up of these non-zero (multiples of unitary) sub-matrices. We have a partition of L. Using these paths we associate with Flij and (G1ij ) matrices denoted by pr(F1ij) and pr(G1ij) respectively where pr(F1ij) and pr(G1ij) are multiples of unitary. If there exist pr(Flij) which are not multiples of identity then we diagonalize these matrices and move a step closer to the simple case and the given collection is said to be in Reduction-form. If not, the collection is in Solution-form. In Solution-form we identify a unitary matrix U sol = blk −diag(U1sol , U2 sol , …, Ur sol )where U isol is a pi ×pi unitary matrix that is expressed in terms of Ud(i) by using the path from i to[d(i)]( i Є [d(i)], d(i) Є L, Ud(i) is free). We show that there exists U such that U HlU∗ = Dl if and only if pr((Flij) = pr(G1ij) and U solHlU sol∗ = Dl. Thus in a maximum of n steps the algorithm solves the Simultaneous Unitary Similarity problem for hermitian matrices. In the second chapter we also relate the Simultaneous Unitary Similarity problem for hermitian matrices to the simultaneous closed system evolution problem for quantum states. In the third chapter we describe algorithms to solve the Unitary Similarity problem for square matrices (single ordered pair) and the Simultaneous Unitary Equivalence problem for rectangular matrices. These problems are related to the Simultaneous Unitary Similarity problem for hermitian matrices. The algorithms described in this chapter are similar in flow to the algorithm described in the second chapter. This shows that it is the fact that we are looking for unitary similarity that makes these forms possible. The hermitian (or normal)nature of the matrices is of secondary importance. Non-reductive-form is the same as in the hermitian case. The definition of the paths changes a little. But once the paths are defined and the set L is partitioned the definitions of Reduction-form and Solution-form are similar to their counterparts in the hermitian case. In the fourth chapter we analyze the worst case complexity of the proposed algorithms. The main computation in all these algorithms is that of diagonalizing normal matrices, partitioning L and calculating the products pr((Flij) = pr(G1ij). Finding the partition of L is like partitioning an undirected graph in the square case and partitioning a bi-graph in the rectangular case. Also, in this chapter we demonstrate the working of the proposed algorithms by running through the steps of the algorithms for three examples. In the fifth and the final chapter we show that finding if a given collection of ordered pairs of normal matrices is Simultaneously Similar is same as finding if the collection is Simultaneously Unitarily Similar. We also discuss why an algorithm to solve the Simultaneous Similarity problem, along the lines of the algorithms we have discussed in this thesis, may not exist. (For equations pl refer the pdf file)