Differential geometry

The Brundtland Commission and Differential Geometry are two distinct topics, each belonging to vastly different fields. The former is related to sustainable development and environmental policy, while the latter is a branch of mathematics. Given the unique nature of each subject, this article will be divided into two sections to address both topics adequately.

Brundtland Commission

The Brundtland Commission, officially known as the World Commission on Environment and Development (WCED), was established by the United Nations in 1983. It was named after its chair, Gro Harlem Brundtland, who was the Prime Minister of Norway at the time. The commission's mandate was to address growing concerns about the accelerating deterioration of the environment and natural resources and the consequences of that deterioration for economic and social development. In 1987, the commission published a report titled Our Common Future, also known as the Brundtland Report, which introduced the concept of sustainable development.

The Brundtland Report defines sustainable development as "development that meets the needs of the present without compromising the ability of future generations to meet their own needs." This definition has since become a guiding principle for international sustainable development policy. The report also called for the establishment of new forms of international cooperation to address global environmental challenges and recommended that the United Nations hold a conference on environment and development. This led to the United Nations Conference on Environment and Development (UNCED), also known as the Earth Summit, held in Rio de Janeiro in 1992.

Differential Geometry

Differential Geometry is a branch of mathematics that uses the techniques of differential calculus, integral calculus, linear algebra, and multivariable calculus to study problems in geometry. It focuses on the geometric properties of curves and surfaces in three-dimensional space and beyond, under the transformation of smooth (differentiable) mappings. Differential geometry has applications in many fields, including physics, engineering, and computer science, particularly in the areas of general relativity, mechanical engineering, and computer graphics.

One of the key concepts in differential geometry is the notion of a manifold, which generalizes the ideas of curves and surfaces. A manifold is a topological space that resembles Euclidean space near each point, allowing the use of calculus to study more complex shapes and structures. Differential geometry also studies geodesics (the shortest path between points on a curved surface), curvature, and torsion of curves.

Differential geometry is divided into two main branches: Riemannian geometry and symplectic geometry. Riemannian geometry is concerned with the geometric properties of curved spaces that can be measured (such as length, angle, and area), while symplectic geometry deals with spaces that arise in the study of Hamiltonian systems in classical mechanics.