To measure distances and angles on manifolds, the manifold must be Riemannian. Legendrian submanifolds are analogous to Lagrangian submanifolds of symplectic manifolds. Tangent spaces of differentiable manifolds are Euclidean vector spaces. There are similar isomorphisms on symplectic manifolds. Differential topology is the field dealing with differentiable functions on differentiable manifolds. The theorem is foundational in differential topology and calculus on manifolds. Riemannian manifolds and Riemann surfaces are named after Riemann. Critical points are fundamental for studying the topology of manifolds and real algebraic varieties. Fuchs equations of the corresponding mirror manifolds. It is possible to develop a calculus for differentiable manifolds. This generalizes the notion of geodesic for Riemannian manifolds. The d'Alembert operator generalizes to a hyperbolic operator on pseudo-Riemannian manifolds. Eikonal solvers on triangulated manifolds are. They admit generalizations to surfaces embedded in more general Riemannian manifolds. It is common to place additional requirements on topological manifolds.