Related papers: Geometry of geodesics
A complete treatment of the intersections of two geodesics on the surface of an ellipsoid of revolution is given. With a suitable metric for the distances between intersections, bounds are placed on their spacing. This leads to fast and…
Over the past six years, a detailed framework has been constructed to unravel the quantum nature of the Riemannian geometry of physical space. A review of these developments is presented at a level which should be accessible to graduate…
These lecture notes provide an introduction to logarithmic geometry with a view towards recent applications in the desingularization theory.
This is a draft of a textbook on differential forms. The primary target audience is sophmore level undergraduates enrolled in what would traditionally be a course in vector calculus. Later chapters will be of interest to advaced…
These are notes for a Ph.D.\ course I held at SISSA, Trieste, in the Winter 2025. We review well-known topics in Riemannian geometry where Lie groups play a fundamental role. Part of the theory of compact connected Lie groups, their…
This book is a regular textbook of analytical geometry covering vector algebra and its applications to describing straight lines, planes, and quadrics in two and three dimensions. The stress is made on vector algebra by using skew-angular…
The geodesic equations are integrated for the Lewis metric and the effects of the different parameters appearing in the Weyl class on the motion of test particles are brought out. Particular attention deserves the appearance of a force…
These notes represent approximately one semester's worth of lectures on introductory general relativity for beginning graduate students in physics. Topics include manifolds, Riemannian geometry, Einstein's equations, and three applications:…
Geometrical aspects of quantum computing are reviewed elementarily for non-experts and/or graduate students who are interested in both Geometry and Quantum Computation. In the first half we show how to treat Grassmann manifolds which are…
We overview some of the foundations of the so-called henselian rigid geometry, and show that henselian rigid geometry has many aspects, useful in applications, that one cannot expect in the usual rigid geometry. This is done by announcing a…
The ``geometry'', in the sense of the classical differential geometry of smooth manifolds (CDG), is put under scrutiny from the point of view of Abstract Differential Geometry (ADG), along with resulting, thereby, potential physical…
Nonlinear analysis has played a prominent role in the recent developments in geometry and topology. The study of the Yang-Mills equation and its cousins gave rise to the Donaldson invariants and more recently, the Seiberg-Witten invariants.…
We collect together various facts about G_2 and Spin(7) geometry which are likely well known but which do not seem to have appeared explicitly in the literature before. These notes should be useful to graduate students and new researchers…
This is a short presentation of some classical results on finite dimensional complex Lie algebras (classification of nilpotent Lie algebras, deformations and perturbations, contractions and rigidity). We present some applications to…
We present an overview of the differential geometry of curves and surfaces using examples from soft matter as illustrations. The presentation requires a background only in vector calculus and is otherwise self-contained.
We review the key mathematical concepts necessary for studying Geometric Deep Learning.
A new field of discrete differential geometry is presently emerging on the border between differential and discrete geometry. Whereas classical differential geometry investigates smooth geometric shapes (such as surfaces), and discrete…
I give a short review of the theory of twisted symmetries of differential equations, emphasizing geometrical aspects. Some open problems are also mentioned.
Euclidean distance geometry is the study of Euclidean geometry based on the concept of distance. This is useful in several applications where the input data consists of an incomplete set of distances, and the output is a set of points in…
Circular and radial geodesics are studied in the spacetime described by the $\gamma$ metric. Their behaviour is compared with the spherically symmetric situation, bringing out the sensitivity of the trajectories to deviations from spherical…