15 Facts About Differential geometry

Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds.

Since the late 19th century, differential geometry has grown into a field concerned more generally with geometric structures on differentiable manifolds.

Differential geometry is closely related to, and is sometimes taken to include, differential topology, which concerns itself with properties of differentiable manifolds which do not rely on any additional geometric structure .

Differential geometry is related to the geometric aspects of the theory of differential equations, otherwise known as geometric analysis.

Differential geometry finds applications throughout mathematics and the natural sciences.

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Outside of physics, differential geometry finds applications in chemistry, economics, engineering, control theory, computer graphics and computer vision, and recently in machine learning.

Field of differential geometry became an area of study considered in its own right, distinct from the more broad idea of analytic geometry, in the 1800s, primarily through the foundational work of Carl Friedrich Gauss and Bernhard Riemann, and in the important contributions of Nikolai Lobachevsky on hyperbolic geometry and non-Euclidean geometry and throughout the same period the development of projective geometry.

Concrete models of hyperbolic Differential geometry were produced by Eugenio Beltrami later in the 1860s, and Felix Klein coined the term non-Euclidean Differential geometry in 1871, and through the Erlangen program put Euclidean and non-Euclidean geometries on the same footing.

In 1899 Luigi Bianchi produced his Lectures on differential geometry which studied differential geometry from Riemann's perspective, and a year later Tullio Levi-Civita and Gregorio Ricci-Curbastro produced their textbook systematically developing the theory of absolute differential calculus and tensor calculus.

Subject of modern differential geometry emerged out of the early 1900s in response to the foundational contributions of many mathematicians, including importantly the work of Henri Poincare on the foundations of topology.

CR Differential geometry is the study of the intrinsic Differential geometry of boundaries of domains in complex manifolds.

Conformal Differential geometry is the study of the set of angle-preserving transformations on a space.

Differential geometry topology is the study of global geometric invariants without a metric or symplectic form.

Differential geometry topology starts from the natural operations such as Lie derivative of natural vector bundles and de Rham differential of forms.

Loosely speaking, this structure by itself is sufficient only for developing analysis on the manifold, while doing geometry requires, in addition, some way to relate the tangent spaces at different points, i e a notion of parallel transport.