Riemannian Geometry


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Chapter 1: Foundations.

Current students. Co-requisite modules None. Prohibited combinations None.

Riemannian Geometry

Sitemap Studying Manage your studies Assessment and examination Develop your skills Graduation University card Work, volunteering and career planning Get ideas Look for work Get skills and experience If things go wrong. Website feedback Online Form - Staff and student feedback. In particular, the problems of cross-session and cross-subject classification are considered.

What's a Tensor?

These problems concern the ability to use data from previous sessions or from a database of past users to calibrate and initialize the classifier, allowing a calibration-less BCI mode of operation. Methods: Data are represented using spatial covariance matrices of the EEG signals, exploiting the recent successful techniques based on the Riemannian geometry of the manifold of symmetric positive definite SPD matrices. Cross-session and cross-subject classification can be difficult, due to the many changes intervening between sessions and between subjects, including physiological, environmental, as well as instrumental changes.

Then, classification is performed both using a standard minimum distance to mean classifier, and through a probabilistic classifier recently developed in the literature, based on a density function mixture of Riemannian Gaussian distributions defined on the SPD manifold.


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Results: The improvements in terms of classification performances achieved by introducing the affine transformation are documented with the analysis of two BCI datasets. HW 4: 1. Petersen, p. Also compute how the Ricci curvature changes under the rescaling of the metric.


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Familiarize yourself with Petersen, pp. October 2 Curvature computations: product of spheres, rotationally symmetric metrics. October 7 Curvature of rotationally symmetric metrics.

Semi-Riemannian Geometry With Applications to Relativity, Volume - 1st Edition

The Hopf fibration. HW 6 October 16 Spheres as warped products. Joins of topological spaces. The Hopf fibration continued. October 20, pm, Location: Krieger note special time Complex projective space and the generalized Hopf fibration.

Riemannian Geometry I

October 21 Curvature of the complex projective plane. HW 7 November 4 Basics of hypersurface theory. The Berger spheres and the complex projective plane are not hypersurfaces. November 6 Parallel transport.

Journal of Differential Geometry

Parallel transport on the 2-sphere. The metric space structure of a Riemannian manifold. November 13 Segments.

First variation for geodesics. Characterization of segments. HW 9 November 18 Jacobi theory: Jacobi fields as critical points for the Lagrangian coming from the second variation of energy. November 20 Jacobi theory: minimal fields, Jacobi variations.

Submission history

November 25 Jacobi fields: the vector space of Jacobi fields, normal Jacobi fields and the Jacobi theorem on the characterization of minimizing geodeiscs via conjugate points. December 2 The Gauss Lemma and some consequences. December 4 Normal and geodesic coordinates. On the characterization of the cut-locus. HW 10 December 9, 10am, Krieger note the special time and location Student presentations.

Riemannian Geometry Riemannian Geometry
Riemannian Geometry Riemannian Geometry
Riemannian Geometry Riemannian Geometry
Riemannian Geometry Riemannian Geometry
Riemannian Geometry Riemannian Geometry
Riemannian Geometry Riemannian Geometry

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