Geometric analysis is a mathematical

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

 discipline at the interface of differential geometry and differential equations. It includes both the use of geometrical methods in the study of partial differential equation

Partial differential equation

In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...

s (when it is also known as "geometric PDE"), and the application of the theory of partial differential equations to geometry. It incorporates problems involving curves and surfaces, or domains with curved boundaries, but also the study of Riemannian manifold

Riemannian manifold

In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...

s in arbitrary dimension. The calculus of variations

Calculus of variations

Calculus of variations is a field of mathematics that deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. A functional is usually a mapping from a set of functions to the real numbers. Functionals are often formed as definite integrals involving unknown...

 is sometimes regarded as part of geometric analysis, because differential equations arising from variational principle

Variational principle

A variational principle is a scientific principle used within the calculus of variations, which develops general methods for finding functions which minimize or maximize the value of quantities that depend upon those functions...

s have a strong geometric content. Geometric analysis also includes global analysis, which concerns the study of differential equations on manifolds, and the relationship between differential equations and topology

Topology

Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...

.