A report on Differential geometry

A triangle immersed in a saddle-shape plane (a hyperbolic paraboloid), as well as two diverging ultraparallel lines.
An osculating circle

Mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds.

- Differential geometry
A triangle immersed in a saddle-shape plane (a hyperbolic paraboloid), as well as two diverging ultraparallel lines.

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Overall
An illustration of Desargues' theorem, a result in Euclidean and projective geometry

Geometry

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

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

An illustration of Desargues' theorem, a result in Euclidean and projective geometry
A European and an Arab practicing geometry in the 15th century
Woman teaching geometry. Illustration at the beginning of a medieval translation of Euclid's Elements, (c. 1310).
An illustration of Euclid's parallel postulate
Acute (a), obtuse (b), and straight (c) angles. The acute and obtuse angles are also known as oblique angles.
A sphere is a surface that can be defined parametrically (by or implicitly (by x2 + y2 + z2 − r2 = 0.)
Visual checking of the Pythagorean theorem for the (3, 4, 5) triangle as in the Zhoubi Suanjing 500–200 BC. The Pythagorean theorem is a consequence of the Euclidean metric.
The Koch snowflake, with fractal dimension=log4/log3 and topological dimension=1
A tiling of the hyperbolic plane
Differential geometry uses tools from calculus to study problems involving curvature.
A thickening of the trefoil knot
Quintic Calabi–Yau threefold
Discrete geometry includes the study of various sphere packings.
The Cayley graph of the free group on two generators a and b
Bou Inania Madrasa, Fes, Morocco, zellige mosaic tiles forming elaborate geometric tessellations
The Pythagoreans discovered that the sides of a triangle could have incommensurable lengths.

Since then, the scope of geometry has been greatly expanded, and the field has been split in many subfields that depend on the underlying methods—differential geometry, algebraic geometry, computational geometry, algebraic topology, discrete geometry (also known as combinatorial geometry), etc.—or on the properties of Euclidean spaces that are disregarded—projective geometry that consider only alignment of points but not distance and parallelism, affine geometry that omits the concept of angle and distance, finite geometry that omits continuity, and others.

A nondifferentiable atlas of charts for the globe. The results of calculus may not be compatible between charts if the atlas is not differentiable. In the center and right charts, the Tropic of Cancer is a smooth curve, whereas in the left chart it has a sharp corner. The notion of a differentiable manifold refines that of a manifold by requiring the functions that transform between charts to be differentiable.

Differentiable manifold

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Type of manifold that is locally similar enough to a vector space to allow one to apply calculus.

Type of manifold that is locally similar enough to a vector space to allow one to apply calculus.

A nondifferentiable atlas of charts for the globe. The results of calculus may not be compatible between charts if the atlas is not differentiable. In the center and right charts, the Tropic of Cancer is a smooth curve, whereas in the left chart it has a sharp corner. The notion of a differentiable manifold refines that of a manifold by requiring the functions that transform between charts to be differentiable.

This leads to such mathematical machinery as the exterior calculus. The study of calculus on differentiable manifolds is known as differential geometry.

3rd century BC Greek mathematician Euclid (holding calipers), as imagined by Raphael in this detail from The School of Athens (1509–1511)

Mathematics

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Area of knowledge that includes such topics as numbers , formulas and related structures (algebra), shapes and the spaces in which they are contained (geometry), and quantities and their changes (calculus and analysis).

Area of knowledge that includes such topics as numbers , formulas and related structures (algebra), shapes and the spaces in which they are contained (geometry), and quantities and their changes (calculus and analysis).

3rd century BC Greek mathematician Euclid (holding calipers), as imagined by Raphael in this detail from The School of Athens (1509–1511)
The distribution of prime numbers is a central point of study in number theory. This Ulam spiral serves to illustrate it, hinting, in particular, at the conditional independence between being prime and being a value of certain quadratic polynomials.
The quadratic formula expresses concisely the solutions of all quadratic equations
Rubik's cube: the study of its possible moves is a concrete application of group theory
The Babylonian mathematical tablet Plimpton 322, dated to 1800 BC.
Archimedes used the method of exhaustion, depicted here, to approximate the value of pi.
The numerals used in the Bakhshali manuscript, dated between the 2nd century BC and the 2nd century AD.
A page from al-Khwārizmī's Algebra
Leonardo Fibonacci, the Italian mathematician who introduced the Hindu–Arabic numeral system invented between the 1st and 4th centuries by Indian mathematicians, to the Western World.
Leonhard Euler created and popularized much of the mathematical notation used today.
Carl Friedrich Gauss, known as the prince of mathematicians
The front side of the Fields Medal
Euler's identity, which American physicist Richard Feynman once called "the most remarkable formula in mathematics".

Such curves can be defined as graph of functions (whose study led to differential geometry).

Carl Friedrich Gauss in 1828

Differential geometry of surfaces

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Carl Friedrich Gauss in 1828
Bernhard Riemann (1826-1866)
Coordinate changes between different local charts must be smooth
A hyperboloid of two sheets
A torus
A helicoid
A quadric ellipsoid
A single-sheeted quadric hyperboloid which is a ruled surface in two different ways.
Surfaces with (from l. to r.) constant negative, zero and positive Gaussian curvature
Eugenio Beltrami (1835-1899)
Wilhelm Blaschke (1885-1962)
A geodesic triangle on the sphere. The geodesics are great circle arcs.
Carl Jacobi (1804–1851)
Contour lines tracking the motion of points on a fixed curve moving along geodesics towards a basepoint
A triangulation of the torus
A triangle in the plane
A spherical triangle
Henri Poincaré (1854-1912)
Tullio Levi-Civita (1873-1941)
Parallel transport of a vector around a geodesic triangle on the sphere. The length of the transported vector and the angle it makes with each side remain constant.
Élie Cartan in 1904
Shortest loop on a torus

In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric.

The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface.

Manifold

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Topological space that locally resembles Euclidean space near each point.

Topological space that locally resembles Euclidean space near each point.

The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface.
The surface of the Earth requires (at least) two charts to include every point. Here the globe is decomposed into charts around the North and South Poles.
Figure 1: The four charts each map part of the circle to an open interval, and together cover the whole circle.
Four manifolds from algebraic curves: circles, parabola, hyperbola, cubic.
The chart maps the part of the sphere with positive z coordinate to a disc.
A finite cylinder is a manifold with boundary.
Möbius strip
The Klein bottle immersed in three-dimensional space
A Morin surface, an immersion used in sphere eversion

During the 1930s Hassler Whitney and others clarified the foundational aspects of the subject, and thus intuitions dating back to the latter half of the 19th century became precise, and developed through differential geometry and Lie group theory.

Riemannian manifold

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In differential geometry, a Riemannian manifold or Riemannian space (M, g), so called after the German mathematician Bernhard Riemann, is a real, smooth manifold M equipped with a positive-definite inner product gp on the tangent space TpM at each point p.

Bernhard Riemann, c. 1863

Bernhard Riemann

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Bernhard Riemann, c. 1863
Riemann's tombstone in Biganzolo in Piedmont, Italy.

Georg Friedrich Bernhard Riemann ( 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry.

Bernhard Riemann

Riemannian geometry

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Bernhard Riemann

Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point that varies smoothly from point to point.

Tensor

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Components stress tensor.svg (.

Components stress tensor.svg (.

The concept enabled an alternative formulation of the intrinsic differential geometry of a manifold in the form of the Riemann curvature tensor.

According to general relativity, objects in a gravitational field behave similarly to objects within an accelerating enclosure. For example, an observer will see a ball fall the same way in a rocket (left) as it does on Earth (right), provided that the acceleration of the rocket is equal to 9.8 m/s2 (the acceleration due to gravity at the surface of the Earth).

General relativity

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According to general relativity, objects in a gravitational field behave similarly to objects within an accelerating enclosure. For example, an observer will see a ball fall the same way in a rocket (left) as it does on Earth (right), provided that the acceleration of the rocket is equal to 9.8 m/s2 (the acceleration due to gravity at the surface of the Earth).
Light cone
Schematic representation of the gravitational redshift of a light wave escaping from the surface of a massive body
Deflection of light (sent out from the location shown in blue) near a compact body (shown in gray)
Ring of test particles deformed by a passing (linearized, amplified for better visibility) gravitational wave
Newtonian (red) vs. Einsteinian orbit (blue) of a lone planet orbiting a star. The influence of other planets is ignored.
Orbital decay for PSR 1913+16: time shift (in s), tracked over 30 years (2006).
Orbital decay for PSR J0737−3039: time shift (in s), tracked over 16 years (2021).
Einstein cross: four images of the same astronomical object, produced by a gravitational lens
Artist's impression of the space-borne gravitational wave detector LISA
Simulation based on the equations of general relativity: a star collapsing to form a black hole while emitting gravitational waves
This blue horseshoe is a distant galaxy that has been magnified and warped into a nearly complete ring by the strong gravitational pull of the massive foreground luminous red galaxy.
Penrose–Carter diagram of an infinite Minkowski universe
The ergosphere of a rotating black hole, which plays a key role when it comes to extracting energy from such a black hole
Projection of a Calabi–Yau manifold, one of the ways of compactifying the extra dimensions posited by string theory
Simple spin network of the type used in loop quantum gravity
Observation of gravitational waves from binary black hole merger GW150914

General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics.