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Introduction to Differential Geometry

Differential geometry uses the methods of differential calculus and integral calculus, with the multi linear and linear algebra to study problems in geometry. The hypothesis of space and plane curves and of surfaces in the 3-dimensional Euclidean space created the basis for development of differential geometry during the 18th century and 19th century. As in the late 19^{th} century, differential geometry has grown into a field concerned more usually with the geometric structures on differentiable manifolds. The differential geometry is closely related to differential topology and the geometric features of the theory of differential geometry s. The differential geometry of surfaces captures numerous key ideas and methods characteristic of this field.

**Branches of differential geometry:**

**Riemannian geometry:**

The Riemannian geometry studies smooth manifolds that are the Riemannian manifolds with a Riemannian metric. It is a notion of distance expressed by means of a smooth positive definite symmetric bilinear form defined on the tangent space at each point. It generalizes the Euclidean geometry to spaces that are not essentially flat, even though they resemble the Euclidean space at each infinite point that is in the initial order of approximation. Numerous concepts based on length, such as the area of arc length of curves, plane regions and volume of solids all possess natural analogues in Riemannian geometry. The concept of a directional derivative of a function from multivariable calculus is extended in Riemannian geometry to the notion of a covariant derivative of a tensor. Many methods and concepts of analysis and Differential geometry s have been generalized to the setting of Riemannian manifolds.

**Pseudo-Riemannian Geometry:**

The Pseudo-Riemannian geometry generalizes Riemannian geometry to the situation in which the metric tensor do not require to be positive-definite. The special case of this is a Lorentzian manifold that is the mathematical foundation of Einstein's general relativity theory of gravity.

**Finsler Geometry:**

Finsler geometry has the Finsler manifold as the main study object. It is a differential manifold with Finsler metric that is a Banach norm defined on each tangent space. A Finsler metric is a more common structure than a Riemannian metric. The Finsler structure on a manifold M is a function F: TM → [0, ∞) which is as follows:

1.) F(x, my) = |m|F(x,y) for all x, y in TM

2.) F is infinitely differentiable in TM - {0},

3.) The vertical Hessian of F2 is positive definite.

**Symplectic geometry:**

The study of Symplectic manifolds is termed as Symplectic geometry. Almost all Symplectic manifold is a differentiable manifold prepared with a smoothly varying non-degenerate skew-symmetric bilinear form on each tangent space which is a non degenerate 2-form ω which is known as the Symplectic form. The Symplectic manifold is almost Symplectic manifold for which the Symplectic form ω is closed i.e., dω = 0.

**Contact geometry:**

This kind of differential geometry deals with certain manifolds of odd dimension. It is closely associated to Symplectic geometry and like the later; it originated in questions of traditional mechanics.

**CR geometry:**

This kind of differential geometry is the study of the intrinsic geometry of boundaries of domains in the complex manifolds.

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