# A report onGeometry, Mathematics and Manifold

Geometry is, with arithmetic, one of the oldest branches of mathematics.

- Geometry

Mathematics is an area of knowledge that includes such topics as numbers (arithmetic, number theory), formulas and related structures (algebra), shapes and the spaces in which they are contained (geometry), and quantities and their changes (calculus and analysis).

- Mathematics

In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point.

- Manifold

The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described in terms of well-understood topological properties of simpler spaces.

- Manifold

This implies that surfaces can be studied intrinsically, that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry.

- Geometry

Manifold theory, the study of shapes that are not necessarily embedded in a larger space

- Mathematics

## Euclidean space

Euclidean space is the fundamental space of geometry, intended to represent physical space.

Originally, that is, in Euclid's Elements, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer dimension, including the three-dimensional space and the Euclidean plane (dimension two).

This way of defining coordinates extends easily to other mathematical structures, and in particular to manifolds.

## Topological space

In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance.

Common types of topological spaces include Euclidean spaces, metric spaces and manifolds.

## Differential geometry

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

It is intimately linked to the development of geometry more generally, of the notion of space and shape, and of topology, especially the study of manifolds.

## Linear algebra

Linear algebra is the branch of mathematics concerning linear equations such as:

For instance, linear algebra is fundamental in modern presentations of geometry, including for defining basic objects such as lines, planes and rotations.

Linear algebra is flat differential geometry and serves in tangent spaces to manifolds.

## Plane (geometry)

In mathematics, a plane is a flat, two-dimensional surface that extends indefinitely.

Many fundamental tasks in mathematics, geometry, trigonometry, graph theory, and graphing are performed in a two-dimensional space, often in the plane.

Differential geometry views a plane as a 2-dimensional real manifold, a topological plane which is provided with a differential structure.