Developing a Headset Integrating Vision Testing and VR with EEG Scan

Consider a Riemannian surface with positive Gaussian curvature, composed of a cylinder and two hemispheres at each end. Since it is established that surfaces of negative curvature exhibit hyperbolicity, it is natural to ask if we get similar results for surfaces of positive curvature. If our surface is flattened in one direction, and geodesic flow is applied, we want to show that this system exhibits characteristics of hyperbolicity. More precisely, we will show that the Lyapunov exponent is positive. We will first discretize geodesics by using nested functions that mimic an iterative map. This approximation is inspired by Euler's fixed step method in which the parameters will be the step size and number of steps. In order to iterate the geodesic, we will fix the step size and create a direction vector parallel to our initial tangent vector. As our direction vector leaves the surface, we need to project our point back onto the surface and our vector back onto the tangent space. We will use methods of Lagrange multipliers and the Gram-Schmidt Method for these respective projections. Once we approximate our geodesic flow, we can determine the sign of our Lyapunov exponent using methods of linear regression. Since this is a numerical approximation, there will be issues of uncertainty, truncating, and rounding due to the floating point problem in programming. However, this project will introduce new mechanisms to produce chaotic behavior and can be extended to higher dimensions.