**Differential geometry** is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The theory of plane and space curves and surfaces in the three-dimensional Euclidean space formed the basis for development of differential geometry during the 18th century and the 19th century.

Since the late 19th century, differential geometry has grown into a field concerned more generally with the geometric structures on differentiable manifolds. Differential geometry is closely related to differential topology and the geometric aspects of the theory of differential equations. The differential geometry of surfaces captures many of the key ideas and techniques endemic to this field.

*Complex differential geometry* is the study of complex manifolds.
An almost complex manifold is a *real* manifold $$*Complex differential geometry* is the study of complex manifolds.
An almost complex manifold is a *real* manifold $M$, endowed with a tensor of type (1, 1), i.e. a vector bundle endomorphism (called an *almost complex structure*)

- $J:T$
It follows from this definition that an almost complex manifold is even-dimensional.

An almost complex manifold is called

*complex*if $N_{J}=0$, where $N_{J}$ is a tensor of type (2, 1) related to $N_{J}=0$, where $N_{J}$ is a tensor of type (2, 1) related to $J$, called the Nijenhuis tensor (or sometimes the*torsion*). An almost complex manifold is complex if and only if it admits a holomorphic coordinate atlas. An*almost Hermitian structure*is given by an almost complex structure*J*, along with a Riemannian metric*g*, satisfying the compatibility conditionAn almost Hermitian structure defines naturally a differential two-form

- $\omega _{J,g}(X,Y):=g(JX,Y)\,$$N_{J}=0{\mbox{ and }}d\omega =0\,$
- $\nabla$ is the Levi-Civita connection of $g$. In this case, $(J,g)$ is called a
*Kähler structure*, and a*Kähler manifold*is a manifold endowed with a Kähler structure. In particular, a Kähler manifold is both a complex and a symplectic manifold. A large class of Kähler manifolds (the class of Hodge manifolds) is given by all the smooth complex projective varieties.### CR geometry

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

### Differential topology

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

Differential topology starts from the natural operations such as Lie derivative of natural vector bundles and de Rham differential of forms. Beside Lie algebroids, also Courant algebroids start playing a more important role.

### Lie groups

A Lie group is a CR geometry is the study of the intrinsic geometry of boundaries of domains in complex manifolds.

### Differential topology

Gauge theory is the study of connections on vector bundles and principal bundles, and arises out of problems in mathematical physics and physical gauge theories which underpin the standard model of particle physics. Gauge theory is concerned with the study of differential equations for connections on bundles, and the resulting geometric moduli spaces of solutions to these equations as well as the invariants that may be derived from them. These equations often arise as the Euler–Lagrange equations describing the equations of motion of certain physical systems in quantum field theory, and so their study is of considerable interest in physics.

## Bundles and connections

The apparatus of vector bundles, princ

The apparatus of vector bundles, principal bundles, and connections on bundles plays an extraordinarily important role in modern differential geometry. A smooth manifold always carries a natural vector bundle, the tangent bundle. 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. An important example is provided by affine connections. For a surface in

**R**^{3}, tangent planes at different points can be identified using a natural path-wise parallelism induced by the ambient Euclidean space, which has a well-known standard definition of metric and parallelism. In Riemannian geometry, the Levi-Civita connection serves a similar purpose. (The Levi-Civita connection defines path-wise parallelism in terms of a given arbitrary Riemannian metric on a manifold.) More generally, differential geometers consider spaces with a vector bundle and an arbitrary affine connection which is not defined in terms of a metric. In physics, the manifold may be the space-time continuum and the bundles and connections are related to various physical fields.## Intrinsic versus extrinsicFrom the beginning and through the middle of the 18th century, differential geometry was studied from the

*extrinsic*point of view: curves and surfaces were considered as lying in a Euclidean space of higher dimension (for example a surface in an ambient space of three dimensions). The simplest results are those in the differential geometry of curves and differential geometry of surfaces. Starting with the work of Riemann, the*intrinsic*point of view was developed, in which one cannot speak of moving "outside" the geometric object because it is considered to be given in a free-standing way. The fundamental result here is Gauss's theorema egregium, to the effect that Gaussian curvature is an intrinsic invariant.The intrinsic point of view is more flexible. For example, it is useful in relativity where space-time cannot naturally be taken as extrinsic (what would be "outside" of the universe?). However, there is a price to pay in technical complexity: the intrinsic definitions of curvature and curvature and connections become much less visually intuitive.

These two points of view can be reconciled, i.e. the extrinsic geometry can be considered as a structure additional to the intrinsic one. (See the Nash embedding theorem.) In the formalism of geometric calculus both extrinsic and intrinsic geometry of a manifold can be characterized by a single bivector-valued one-form called the shape operator.

^{[5]}Below are some examples of how differential geometry is applied to other fields of science and mathematics.

- In physics, differential geometry has many applications, including:
- Differential geometry is the language in which Albert Einstein's general theory of relativity is expressed. According to the theory, the universe is a smooth manifold equipped with a pseudo-Riemannian metric, which describes the curvature of spacetime. Understanding this curvature is essential for the positioning of satellites into orbit around the earth. Differential geometry is also indispensable in the study of gravitational lensing and black holes.
- Differential forms are used in the study of electromagnetism.
- Differential geometry has applications to both Lagrangian mechanics and Hamiltonian mechanics. Symplectic manifolds in particular can be used to study Hamiltonian systems.
- Riemannian geometry and contact geometry have been used to construct the formalism of geometrothermodynamics which has found applications in classical equilibrium thermodynamics.

- In chemistry and biophysics when modelling cell membrane structure under varying pressure.
- In economics, differential geometry has applications to the field of econometrics.
^{[6]} - Geometric modeling (including computer graphics) and computer-aided geometric design draw on ideas from differential geometry.
- In engineering, differential geometry can be applied to solve problems in digital signal processing.
^{[7]} - In control theory, differential geometry can be used to analyze nonlinear controllers, particularly geometric control
^{[8]} - In probability, statistics, and information theory, one can interpret various structures as Riemannian manifolds, which yields the field of information geometry, particularly via the Fisher information metric.
- In structural geology, differential geometry is used to analyze and describe geologic structures.
- In computer vision, differential geometry is used to analyze shapes.
^{[9]} - In image processing, differential geometry is used to process and analyse data on non-flat surfaces.
^{[10]} - Grigori Perelman's proof of the Poincaré conjecture using the techniques of Ricci flows demonstrated the power of the differential-geometric approach to questions in topology and it highlighted the important role played by its analytic methods.
- In

- $\nabla$ is the Levi-Civita connection of $g$. In this case, $(J,g)$ is called a

- $\omega _{J,g}(X,Y):=g(JX,Y)\,$$N_{J}=0{\mbox{ and }}d\omega =0\,$