Competencies: Mathematics > Differential Geometry

Curves & Surfaces

Parametric curves, arc length, curvature, torsion, Frenet-Serret frame. Regular surfaces, coordinate patches, tangent planes.

High

ML Engineer, Physicist, Robotics Engineer

First & Second Fundamental Forms

Metric tensor on surfaces (I), shape operator and normal curvature (II), principal curvatures from eigenvalues of shape operator.

High

Physicist, Robotics Engineer

Gaussian Curvature

Product of principal curvatures, Gauss’s Theorema Egregium (intrinsic invariant), positive/negative/zero curvature surfaces.

Critical

Physicist, ML Engineer, Robotics Engineer

Geodesics

Shortest paths on surfaces, geodesic equations, exponential map, geodesic completeness, Hopf-Rinow theorem.

High

Physicist, Robotics Engineer, ML Engineer

Smooth Manifolds

Charts, atlases, smooth structure, smooth maps between manifolds, examples (spheres, projective spaces, Lie groups).

Critical

Physicist, ML Engineer, Research Mathematician

Tangent Spaces

Tangent vectors as derivations, tangent bundle, pushforward (differential) of smooth maps, vector fields.

High

Physicist, ML Engineer

Differential Forms

Covectors, exterior algebra, wedge product, pullback, exterior derivative, Stokes' theorem on manifolds.

High

Physicist, ML Engineer, Research Mathematician

Riemannian Metrics

Inner product on tangent spaces, length of curves, volume form, isometries, Riemannian manifolds as metric spaces.

Critical

Physicist, ML Engineer

Connections & Parallel Transport

Levi-Civita connection, Christoffel symbols, covariant derivative, parallel transport along curves, holonomy.

High

Physicist, Research Mathematician

Curvature Tensor

Riemann curvature tensor, Ricci curvature, scalar curvature, sectional curvature, Einstein field equations context.

High

Physicist, Research Mathematician