Academic literature on the topic 'Methods of global Riemannian geometry'

The aim of the first part of this paper is to make some reflections on the role of geometrical and topological concepts in the developments of theoretical physics, especially in gauge theory and string theory, and we show the great significance of these concepts for a better understanding of the dynamics of physics. We will claim that physical phenomena essentially emerge from the geometrical and topological structure of space–time. The attempts to solve one of the central problems in 20th theoretical physics, i.e. how to combine gravity and the other forces into an unitary theoretical explanation of the physical world, essentially depends on the possibility of building a new geometrical framework conceptually richer than Riemannian geometry. In fact, it still plays a fundamental role in non-Abelian gauge theories and in superstring theory, thanks to which a great variety of new mathematical structures has emerged. The scope of this presentation is to highlight the importance of these mathematical structures for theoretical physics. A very interesting hypothesis is that the global topological properties of the manifold's model of space–time play a major role in quantum field theory (QFT) and that, consequently, several physical quantum effects arise from the nonlocal changing metrical and topological structure of these manifold. Thus the unification of general relativity and quantum theory require some fundamental breakthrough in our understanding of the relationship between space–time and quantum process. In particular the superstring theories lead to the guess that the usual structure of space–time at the quantum scale must be dropped out from physical thought. Non-Abelian gauge theories satisfy the basic physical requirements pertaining to the symmetries of particle physics because they are geometric in character. They profoundly elucidate the fundamental role played by bundles, connections, and curvature in explaining the essential laws of nature. Kaluza–Klein theories and more remarkably superstring theory showed that space–time symmetries and internal (quantum) symmetries might be unified through the introduction of new structures of space with a different topology. This essentially means, in our view, that "hidden" symmetries of fundamental physics can be related to the phenomenon of topological change of certain class of (presumably) nonsmooth manifolds. In the second part of this paper, we address the subject of topological quantum field theories (TQFTs), which constitute a remarkably important meeting ground for physicists and mathematicians. TQFTs can be used as a powerful tool to probe geometry and topology in low dimensions. Chern–Simons theories, which are examples of such field theories, provide a field theoretic framework for the study of knots and links in three dimensions. These are rare examples of QFTs which can be exactly (nonperturbatively) and explicitly solved. Abelian Chern–Simons theory provides a field theoretic interpretation of the linking and self-linking numbers of a link (i.e. the union of a finite number of disjoint knots). In non-Abelian theories, vacuum expectation values of Wilson link operators yield a class of polynomial link invariants; the simplest of them is the well-known Jones polynomial. Powerful methods for complete analytical and nonperturbative computation of these knot and link invariants have been developed. From these invariants for unoriented and framed links in S3, an invariant for any three-manifold can be easily constructed by exploiting the Lickorish–Wallace surgery presentation of three-manifolds. This invariant up to a normalization is the partition function of the Chern–Simons field theory. Even perturbative analysis of Chern–Simons theories are rich in their mathematical structure; these provide a field theoretic interpretation of Vassiliev knot invariants. In Donaldson–Witten theory perturbative methods have proved their relations to Donaldson invariants. Nonperturbative methods have been applied after the work by Seiberg and Witten on N = 2 supersymmetric Yang–Mills theory. The outcome of this application is a totally unexpected relation between Donaldson invariants and a new set of topological invariants called Seiberg–Witten invariants. Not only in mathematics, Chern–Simons theories find important applications in three- and four-dimensional quantum gravity also. Work on TQFT suggests that a quantum gravity theory can be formulated in three-dimensional space–time. Attempts have been made in the last years to formulate a theory of quantum gravity in four-dimensional space–time using "spin networks" and "spin foams". More generally, the developments of TQFTs represent a sort of renaissance in the relation between geometry and physics. The most important (new) feature of present developments is that links are being established between quantum physics and topology. Maybe this link essentially rests on the fact that both quantum theory and topology are characterized by discrete phenomena emerging from a continuous background. One very interesting example is the super-symmetric quantum mechanics theory, which has a deep geometric meaning. In the Witten super-symmetric quantum mechanics theory, where the Hamiltonian is just the Hodge–Laplacian (whereas the quantum Hamiltonian corresponding to a classical particle moving on a Riemannian manifold is just the Laplace–Beltrami differential operator), differential forms are bosons or fermions depending on the parity of their degrees. Witten went to introduce a modified Hodge–Laplacian, depending on a real-valued function f. He was then able to derive the Morse theory (relating critical points of f to the Betti numbers of the manifold) by using the standard limiting procedures relating the quantum and classical theories. Super-symmetric QFTs essentially should be viewed as the differential geometry of certain infinite-dimensional manifolds, including the associated analysis (e.g. Hodge theory) and topology (e.g. Betti numbers). A further comment is that the QFTs of interest are inherently nonlinear, but the nonlinearities have a natural origin, e.g. coming from non-Abelian Lie groups. Moreover there is usually some scaling or coupling parameter in the theory which in the limit relates to the classical theory. Fundamental topological aspects of such a quantum theory should be independent of the parameters and it is therefore reasonable to expect them to be computable (in some sense) by examining the classical limit. This means that such topological information is essentially robust and should be independent of the fine analytical details (and difficulties) of the full quantum theory. In the last decade much effort has been done to use these QFTs as a conceptual tool to suggest new mathematical results. In particular, they have led to spectacular progress in our understanding of geometry in low dimensions. It is most likely no accident that the usual QFTs can only be renormalized in (space–time) dimensions ≤4, and this is precisely the range in which difficult phenomena arise leading to deep and beautiful theories (e.g. the work of Thurston in three dimensions and Donaldson in four dimensions). It now seems clear that the way to investigate the subtleties of low-dimensional manifolds is to associate to them suitable infinite-dimensional manifolds (e.g. spaces of connections) and to study these by standard linear methods (homology, etc.). In other words we use QFT as a refined tool to study low-dimensional manifolds.

AbstractWe begin a study of invariant isometric immersions into Riemannian manifolds (M, g) equipped with a Riemannian flow generated by a unit Killing vector field ξ. We focus our attention on those (M, g) where ξ is complete and such that the reflections with respect to the flow lines are global isometries (that is, (M, g) is a Killing-transversally symmetric space) and on the subclass of normal flow space forms. General results are derived and several examples are provided.

We study new classes of generic off-diagonal and diagonal cosmological solutions for effective Einstein equations in modified gravity theories (MGTs), with modified dispersion relations (MDRs), and encoding possible violations of (local) Lorentz invariance (LIVs). Such MGTs are constructed for actions and Lagrange densities with two non-Riemannian volume forms (similar to two measure theories (TMTs)) and associated bimetric and/or biconnection geometric structures. For conventional nonholonomic 2 + 2 splitting, we can always describe such models in Finsler-like variables, which is important for elaborating geometric methods of constructing exact and parametric solutions. Examples of such Finsler two-measure formulations of general relativity (GR) and MGTs are considered for Lorentz manifolds and their (co) tangent bundles and abbreviated as FTMT. Generic off-diagonal metrics solving gravitational field equations in FTMTs are determined by generating functions, effective sources and integration constants, and characterized by nonholonomic frame torsion effects. By restricting the class of integration functions, we can extract torsionless and/or diagonal configurations and model emergent cosmological theories with square scalar curvature, R2, when the global Weyl-scale symmetry is broken via nonlinear dynamical interactions with nonholonomic constraints. In the physical Einstein–Finsler frame, the constructions involve: (i) nonlinear re-parametrization symmetries of the generating functions and effective sources; (ii) effective potentials for the scalar field with possible two flat regions, which allows for a unified description of locally anisotropic and/or isotropic early universe inflation related to acceleration cosmology and dark energy; (iii) there are “emergent universes” described by off-diagonal and diagonal solutions for certain nonholonomic phases and parametric cosmological evolution resulting in various inflationary phases; (iv) we can reproduce massive gravity effects in two-measure theories. Finally, we study a reconstructing procedure for reproducing off-diagonal FTMT and massive gravity cosmological models as effective Einstein gravity or Einstein–Finsler theories.

The review focuses on the most promising methods for classifying EEG signals for non-invasive BCIs and theoretical approaches for the successful classification of EEG patterns. The paper provides an overview of articles using Riemannian geometry, deep learning methods and various options for preprocessing and "clustering" EEG signals, for example, common-spatial pattern (CSP). Among other approaches, pre-processing of EEG signals using CSP is often used, both offline and online. The combination of CSP, linear discriminant analysis, support vector machine and neural network (BPNN) made it possible to achieve 91% accuracy for binary classification with exoskeleton control as a feedback. There is very little work on the use of Riemannian geometry online and the best accuracy achieved so far for a binary classification problem is 69.3% in the work. At the same time, in offline testing, the average percentage of correct classification in the considered articles for approaches with CSP – 77.5 ± 5.8%, deep learning networks – 81.7 ± 4.7%, Riemannian geometry – 90.2 ± 6.6%. Due to nonlinear transformations, Riemannian geometry-based approaches and complex deep neural networks provide higher accuracy and better extract of useful information from raw EEG recordings rather than linear CSP transformation. However, in real-time setup, not only accuracy is important, but also a minimum time delay. Therefore, approaches using the CSP transformation and Riemannian geometry with a time delay of less than 500 ms may be in the future advantage.