NonRiemannian

Non-Riemannian geometry refers to a branch of differential geometry that generalizes the concept of Riemannian geometry. While Riemannian geometry is based on the notion of a Riemannian metric, which is a positive-definite symmetric bilinear form on the tangent space of a manifold, non-Riemannian geometry allows for more general types of metrics. These can include indefinite metrics, complex metrics, and even metrics that are not symmetric or bilinear.

One of the most well-known examples of non-Riemannian geometry is Finsler geometry, which generalizes Riemannian geometry by allowing the metric to depend on the direction of the tangent vector. This results in a metric that is not necessarily quadratic in the tangent vector. Another example is Lorentzian geometry, which is used in the study of spacetime in general relativity. In Lorentzian geometry, the metric is indefinite, meaning it has both positive and negative eigenvalues.

Non-Riemannian geometries have applications in various fields, including physics, computer vision, and machine learning. In physics, for instance, Lorentzian geometry is crucial for understanding the behavior of particles and fields in spacetime. In computer vision, non-Riemannian metrics can be used to model the geometry of images and videos more accurately. In machine learning, non-Riemannian geometries can be used to define new types of neural networks and optimization algorithms.

Despite their differences, non-Riemannian geometries share many properties with Riemannian geometry. For example, they can both be used to define concepts such as curvature, geodesics, and parallel transport. However, the specific formulas and properties can differ significantly, reflecting the more general nature of the metrics involved.