Typically, the solution set for the partial differential equations is the form of a high-dimensional manifold. In the beginning, geometry was studied to better understand the physical environment that we live in and the practice continues until today. Knowing what is the "geometry" of this manifold can provide an insight into the nature of these solutions and to the real-world phenomenon that is modelled by differential equations, regardless of whether it is economics, physics, engineering or another quantitative science.1 Take for instance the astonishing performance of Einstein’s theories of general relativity. The most common problem with geometry concerns how to "classify" all manifolds belonging to the same kind. It is a pure geometric theory which describes gravitation through the curvature of four-dimensional "spacetime".1

This means that we first determine the types of manifolds we’re interested in, and then determine whether two manifolds can generally be thought of as identical, or "equivalent" or "equivalent" and then attempt to figure out how many different manifolds that are not equivalent to each other exist.1 Geometry goes beyond physical application, and it’s not untrue to claim that geometric ideas and methods have been a part of every area of mathematics. For instance, we could want to study the surface (2-dimensional manifolds) which are located inside the normal 3-dimensional space we can perceive, and we could consider that two of these surfaces are identical if one is able to be "transformed" to the other through either rotation or translation.1 In modern times the primary object of study in the field of geometry is a manifold.

It is the Riemannian surface geometry that are immersed in 3-space. A manifold represents an entity that can be a complex shape overall but at small scales, that it "looks like" regular space with a particular dimension.1 It was traditionally the first subfield of "differential geometry" that was pioneered by mathematic giants like Gauss as well as Riemann during the early 1800’s. For instance, a 1D manifold is an item that is so that even small parts of it appear as a line but in general, it looks more like a curve than straight lines.1 There are currently numerous geometries that are currently being studied.

A 2-dimensional manifold, at smaller scales, looks similar to it’s a (curved) bit of paper. We will only discuss some of them: It has two different directions that we can travel at any given point. Riemannian geometry. For instance, the top that is the Earth is a two-dimensional manifold.1 It involves the analysis of manifolds that are equipped with the extra structure of the Riemannian measurement that is a method to measure the lengths of angles and curves between the tangent vectors. The n-dimensional manifold also appears at a local level like an ordinary n-dimensional area. A Riemannian manifold is curvilinear, and it’s precisely this curvature which causes the rules that govern classic Euclidean geometry, which we are taught in elementary school in elementary school, to differ.1

It does not have to be related to any concept or concept of "physical spaces". For instance the sum of the interior angles of the "triangle" in a curving Riemannian manifold could be greater than or less than p if the curvature is negative or positive or negative, according to.1 In this case it is the velocity and position of N particles within a room is described using 6N independent variables, since each particle requires 3 numbers to define its position , and three more numbers to indicate its velocity. Algebraic geometry.

Thus that what is known as the "configuration space" of this system is a six-dimensional manifold.1 It is an investigation into algebraic variations that are solutions sets of polynomial systems. equations. If, for any reason, this motion was not independent , but rather restricted in some way, the configuration space could be a manifold having a smaller size.

They may be manifolds, however they also have "singular points" where they aren’t "smooth".1 The collection of solutions to a partial differential equations have the structure of a manifold with a high dimension. Since they are algebraically defined and mathematically, there are a variety of tools that can be used from abstract algebra to analyze these, and in turn, many problems in pure algebra can be better understood by rewriting the issue in terms of algebraic geometry.1 Learning this "geometry" of the manifold may provide new insights into the structure of the solutions, as well as to the actual process that is described by differential equations, no matter if it’s engineering, economics, physics or any other quantitative science.

Additionally, it is possible to examine the various types of any field and not only the complex or real numbers.1 One of the most frequent problems for geometry involves trying to "classify" every manifold of one particular kind. Symplectic geometry. In other words, we first choose which types of manifolds are we interested in, then we decide which manifolds could in essence be considered the same or "equivalent" in the final step, and look to determine the number of similar manifolds exist.1 It involves the research of manifolds that are equipped by an extra structure, referred to as the symplectic shape. For instance, we may find ourselves interested in studying the surface (2-dimensional manifolds) that are within the typical 3-dimensional space we see. Symplectic forms are in a sense (that can be made exact) an alternative to the Riemannian measurement and manifolds with a symplectic form have a distinct behavior in comparison to Riemannian manifolds.1

Moreover, we may determine that two surfaces are comparable if they is "transformed" in the opposite direction via the process of rotation or translation. For instance, a well-known theorem by Darboux states that all manifolds that are symplectic are "locally" identical however, globally they could be very different.1 The study is based on the Riemannian geometrical geometry that is encased in 3-space. This is not the case for Riemannian geometry. It is typically the first subfield in "differential geometry" which was developed by mathematic giants like Gauss as well as Riemann around the time of the 1800’s.1 Symplectic manifolds are naturally arising in physical systems derived from classical mechanics.

Nowadays, there are several Geometry subfields being constantly being investigated. They are referred to as "phases space" in the field of physics. In this article, we will discuss only the most important ones: This particular branch of geometry is extremely topological in its nature.1 Riemannian geometry.

Complex geometry. It refers to the research of manifolds adorned with the added structure of the Riemannian metrics which is a measure to determine the length of angles and curves between the tangent vectors. It involves the research of manifolds that local "look like" normal n-dimensional spaces which are constructed using complex numbers, not the actual numbers.1 A Riemannian manifold exhibits curvature and it’s precisely this curvature that causes the basic laws in traditional Euclidean geometry, which we are taught in elementary schools and high school, to be different. Since the analysis of Holomorphic (or complex analytical) functions is much less rigid than that of the actual situation (for instance, not all smooth functions are real-analytical) there are many less "types" of manifolds with complex structures and there has been greater success in (at at the very least, partial) classifications.1

For instance the sum of interior angles of the "triangle" that is on a curving Riemannian manifold may be greater or less than p, depending on whether the curvature is negative or positive and vice versa.