Differential geometry

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hyperbolic paraboloid), as well as two diverging ultraparallel lines

.

plane
Outline History (Timeline)
Branches Euclidean Non-Euclidean Elliptic Spherical Hyperbolic Non-Archimedean geometry Projective Affine Synthetic Analytic Algebraic Arithmetic Diophantine Differential Riemannian Symplectic Discrete differential Complex Finite Discrete/Combinatorial Digital Convex Computational Fractal Incidence Noncommutative geometry Noncommutative algebraic geometry
Concepts Features Dimension Straightedge and compass constructions Angle Curve Diagonal Orthogonality (Perpendicular) Parallel Vertex Congruence Similarity Symmetry
Zero-dimensional Point
One-dimensional Line segment ray Length
Two-dimensional Plane Area Polygon Triangle Altitude Hypotenuse Pythagorean theorem Parallelogram Square Rectangle Rhombus Rhomboid Quadrilateral Trapezoid Kite Circle Diameter Circumference Area
Three-dimensional Volume Cube cuboid Cylinder Dodecahedron Icosahedron Octahedron Pyramid Platonic Solid Sphere Tetrahedron
Four- / other-dimensional Tesseract Hypersphere
Geometers
by name Aida Aryabhata Ahmes Alhazen Apollonius Archimedes Atiyah Baudhayana Bolyai Brahmagupta Cartan Coxeter Descartes Euclid Euler Gauss Gromov Hilbert Huygens Jyeṣṭhadeva Kātyāyana Khayyám Klein Lobachevsky Manava Minkowski Minggatu Pascal Pythagoras Parameshvara Poincaré Riemann Sakabe Sijzi al-Tusi Veblen Virasena Yang Hui al-Yasamin Zhang List of geometers
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Differential geometry is a

, and the study of these shapes formed the basis for development of modern differential geometry during the 18th and 19th centuries.

Since the late 19th century, differential geometry has grown into a field concerned more generally with geometric structures on differentiable manifolds. A geometric structure is one which defines some notion of size, distance, shape, volume, or other rigidifying structure. For example, in Riemannian geometry distances and angles are specified, in symplectic geometry volumes may be computed, in conformal geometry only angles are specified, and in gauge theory certain fields are given over the space. Differential geometry is closely related to, and is sometimes taken to include, differential topology, which concerns itself with properties of differentiable manifolds that do not rely on any additional geometric structure (see that article for more discussion on the distinction between the two subjects). Differential geometry is also related to the geometric aspects of the theory of differential equations, otherwise known as geometric analysis.

Differential geometry finds applications throughout mathematics and the

.

History and development

The history and development of differential geometry as a subject begins at least as far back as classical antiquity. 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. In this section we focus primarily on the history of the application of infinitesimal methods to geometry, and later to the ideas of tangent spaces, and eventually the development of the modern formalism of the subject in terms of tensors and tensor fields.

Classical antiquity until the Renaissance (300 BC – 1600 AD)

The study of differential geometry, or at least the study of the geometry of smooth shapes, can be traced back at least to

of curves, a concept which did not see a rigorous definition in terms of calculus until the 1600s.

Around this time there were only minimal overt applications of the theory of

There was little development in the theory of differential geometry between antiquity and the beginning of the

Gauss

.

After calculus (1600–1800)

The first systematic or rigorous treatment of geometry using the theory of infinitesimals and notions from

on the foundations of calculus, Leibniz notes that the infinitesimal condition ${\displaystyle d^{2}y=0}$ indicates the existence of an inflection point. Shortly after this time the Bernoulli brothers, Jacob and Johann made important early contributions to the use of infinitesimals to study geometry. In lectures by Johann Bernoulli at the time, later collated by L'Hopital into the first textbook on differential calculus, the tangents to plane curves of various types are computed using the condition ${\displaystyle dy=0}$, and similarly points of inflection are calculated.^[1] At this same time the orthogonality between the osculating circles of a plane curve and the tangent directions is realised, and the first analytical formula for the radius of an osculating circle, essentially the first analytical formula for the notion of curvature, is written down.

In the wake of the development of analytic geometry and plane curves,

later studied by Gauss and others.

Around this same time,

.

Later in the 1700s, the new French school led by

Intrinsic geometry and non-Euclidean geometry (1800–1900)

The field of differential geometry became an area of study considered in its own right, distinct from the more broad idea of analytic geometry, in the 1800s, primarily through the foundational work of Carl Friedrich Gauss and Bernhard Riemann, and also in the important contributions of Nikolai Lobachevsky on hyperbolic geometry and non-Euclidean geometry and throughout the same period the development of projective geometry.

Dubbed the single most important work in the history of differential geometry,

in various non-Euclidean geometries on surfaces.

At this time Gauss was already of the opinion that the standard paradigm of Euclidean geometry should be discarded, and was in possession of private manuscripts on non-Euclidean geometry which informed his study of geodesic triangles.^[6][7] Around this same time János Bolyai and Lobachevsky independently discovered hyperbolic geometry and thus demonstrated the existence of consistent geometries outside Euclid's paradigm. Concrete models of hyperbolic geometry were produced by Eugenio Beltrami later in the 1860s, and Felix Klein coined the term non-Euclidean geometry in 1871, and through the Erlangen program put Euclidean and non-Euclidean geometries on the same footing.^[8] Implicitly, the spherical geometry of the Earth that had been studied since antiquity was a non-Euclidean geometry, an elliptic geometry.

The development of intrinsic differential geometry in the language of Gauss was spurred on by his student,

for the first time, and began the systematic study of differential geometry in higher dimensions. This intrinsic point of view in terms of the Riemannian metric, denoted by ${\displaystyle ds^{2}}$ by Riemann, was the development of an idea of Gauss' about the linear element ${\displaystyle ds}$ of a surface. At this time Riemann began to introduce the systematic use of linear algebra and multilinear algebra into the subject, making great use of the theory of quadratic forms in his investigation of metrics and curvature. At this time Riemann did not yet develop the modern notion of a manifold, as even the notion of a topological space had not been encountered, but he did propose that it might be possible to investigate or measure the properties of the metric of spacetime through the analysis of masses within spacetime, linking with the earlier observation of Euler that masses under the effect of no forces would travel along geodesics on surfaces, and predicting Einstein's fundamental observation of the equivalence principle a full 60 years before it appeared in the scientific literature.^[6][4]

In the wake of Riemann's new description, the focus of techniques used to study differential geometry shifted from the ad hoc and extrinsic methods of the study of curves and surfaces to a more systematic approach in terms of

.

Modern differential geometry (1900–2000)

The subject of modern differential geometry emerged from the early 1900s in response to the foundational contributions of many mathematicians, including importantly the work of Henri Poincaré on the foundations of topology.^[12] At the start of the 1900s there was a major movement within mathematics to formalise the foundational aspects of the subject to avoid crises of rigour and accuracy, known as Hilbert's program. As part of this broader movement, the notion of a topological space was distilled in by Felix Hausdorff in 1914, and by 1942 there were many different notions of manifold of a combinatorial and differential-geometric nature.^[12]

Interest in the subject was also focused by the emergence of Einstein's theory of general relativity and the importance of the Einstein Field equations. Einstein's theory popularised the tensor calculus of Ricci and Levi-Civita and introduced the notation ${\displaystyle g}$ for a Riemannian metric, and ${\displaystyle \Gamma }$ for the Christoffel symbols, both coming from G in Gravitation.

Following this early development, many mathematicians contributed to the development of the modern theory, including

.

In the middle and late 20th century differential geometry as a subject expanded in scope and developed links to other areas of mathematics and physics. The development of

Seiberg–Witten invariants

.

Branches

Riemannian geometry

Riemannian geometry studies

. Many concepts of analysis and differential equations have been generalized to the setting of Riemannian manifolds.

A distance-preserving

.

Pseudo-Riemannian geometry

positive-definite

. A special case of this is a .

Finsler geometry

Finsler geometry has Finsler manifolds as the main object of study. This is a differential manifold with a Finsler metric, that is, a

defined on each tangent space. Riemannian manifolds are special cases of the more general Finsler manifolds. A Finsler structure on a manifold M is a function F : TM → [0, ∞) such that:

1. F(x, my) = m F(x, y) for all (x, y) in TM and all m ≥ 0,
2. F is infinitely differentiable in TM ∖ {0},
3. The vertical Hessian of F² is positive definite.

Symplectic geometry

ω, called the symplectic form. A symplectic manifold is an almost symplectic manifold for which the symplectic form ω is closed: dω = 0.

A diffeomorphism between two symplectic manifolds which preserves the symplectic form is called a

.

By contrast with Riemannian geometry, where the

Contact geometry

Contact geometry deals with certain manifolds of odd dimension. It is close to symplectic geometry and like the latter, it originated in questions of classical mechanics. A contact structure on a (2n + 1)-dimensional manifold M is given by a smooth hyperplane field H in the tangent bundle that is as far as possible from being associated with the level sets of a differentiable function on M (the technical term is "completely nonintegrable tangent hyperplane distribution"). Near each point p, a hyperplane distribution is determined by a nowhere vanishing 1-form ${\displaystyle \alpha }$, which is unique up to multiplication by a nowhere vanishing function:

${\displaystyle H_{p}=\ker \alpha _{p}\subset T_{p}M.}$

A local 1-form on M is a contact form if the restriction of its exterior derivative to H is a non-degenerate two-form and thus induces a symplectic structure on H_p at each point. If the distribution H can be defined by a global one-form ${\displaystyle \alpha }$ then this form is contact if and only if the top-dimensional form

${\displaystyle \alpha \wedge (d\alpha )^{n}}$

is a volume form on M, i.e. does not vanish anywhere. A contact analogue of the Darboux theorem holds: all contact structures on an odd-dimensional manifold are locally isomorphic and can be brought to a certain local normal form by a suitable choice of the coordinate system.

Complex and Kähler geometry

Complex differential geometry is the study of

. An almost complex manifold is a real manifold ${\displaystyle M}$, endowed with a )

${\displaystyle J:TM\rightarrow TM}$, such that ${\displaystyle J^{2}=-1.\,}$

It follows from this definition that an almost complex manifold is even-dimensional.

An almost complex manifold is called complex if ${\displaystyle N_{J}=0}$, where ${\displaystyle N_{J}}$ is a tensor of type (2, 1) related to ${\displaystyle J}$, called the

(or sometimes the torsion). An almost complex manifold is complex if and only if it admits a holomorphic coordinate atlas. An

Riemannian metric

g, satisfying the compatibility condition

${\displaystyle g(JX,JY)=g(X,Y).\,}$

An almost Hermitian structure defines naturally a differential two-form

${\displaystyle \omega _{J,g}(X,Y):=g(JX,Y).\,}$

The following two conditions are equivalent:

1. ${\displaystyle N_{J}=0{\mbox{ and }}d\omega =0\,}$
2. ${\displaystyle \nabla J=0\,}$

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

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

.

Conformal geometry

Conformal geometry is the study of the set of angle-preserving (conformal) transformations on a space.

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 group in the category of smooth manifolds. Beside the algebraic properties this enjoys also differential geometric properties. The most obvious construction is that of a Lie algebra which is the tangent space at the unit endowed with the Lie bracket between left-invariant vector fields. Beside the structure theory there is also the wide field of representation theory.

Geometric analysis

Geometric analysis is a mathematical discipline where tools from differential equations, especially elliptic partial differential equations are used to establish new results in differential geometry and differential topology.

Gauge theory

Main article: Gauge theory (mathematics)

Gauge theory is the study of connections on vector bundles and principal bundles, and arises out of problems in

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, 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 R³, 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. 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 spacetime and the bundles and connections are related to various physical fields.

Intrinsic versus extrinsic

From the beginning and through the middle of the 19th 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

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. However, there is a price to pay in technical complexity: the intrinsic definitions of 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

shape operator.^[15]

Applications

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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
    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.^[16]
• computer-aided geometric design
  draw on ideas from differential geometry.
• In engineering, differential geometry can be applied to solve problems in digital signal processing.^[17]
• In control theory, differential geometry can be used to analyze nonlinear controllers, particularly geometric control^[18]
• 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.^[19]
• In
  image processing, differential geometry is used to process and analyse data on non-flat surfaces.^[20]
• 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 wireless communications, Grassmannian manifolds are used for beamforming techniques in multiple antenna systems.^[21]
• In geodesy, for calculating distances and angles on the mean sea level surface of the Earth, modelled by an ellipsoid of revolution.

See also

• Abstract differential geometry
• Affine differential geometry
• Analysis on fractals
• Basic introduction to the mathematics of curved spacetime
• Discrete differential geometry
• Gauss
• Glossary of differential geometry and topology
• Important publications in differential geometry
• Important publications in differential topology
• Integral geometry
• List of differential geometry topics
• Noncommutative geometry
• Projective differential geometry
• Synthetic differential geometry
• Systolic geometry
• Gauge theory (mathematics)

References

1. ^ ^{a b c d e f g} Struik, D. J. "Outline of a History of Differential Geometry: I." Isis, vol. 19, no. 1, 1933, pp. 92–120. JSTOR, www.jstor.org/stable/225188.
2. ^ Clairaut, A.C., 1731. Recherches sur les courbes à double courbure. Nyon.
3. ^ O'Connor, John J.; Robertson, Edmund F., "Leonhard Euler", MacTutor History of Mathematics Archive, University of St Andrews
4. ^ ^{a b c d e f} Spivak, M., 1975. A comprehensive introduction to differential geometry (Vol. 2). Publish or Perish, Incorporated.
5. ^ Gauss, C.F., 1828. Disquisitiones generales circa superficies curvas (Vol. 1). Typis Dieterichianis.
6. ^ ^{a b c d} Struik, D.J. "Outline of a History of Differential Geometry (II)." Isis, vol. 20, no. 1, 1933, pp. 161–191. JSTOR, www.jstor.org/stable/224886
7. ^ O'Connor, John J.; Robertson, Edmund F., "Non-Euclidean Geometry", MacTutor History of Mathematics Archive, University of St Andrews
8. ^ Milnor, John W., (1982) Hyperbolic geometry: The first 150 years, Bull. Amer. Math. Soc. (N.S.) Volume 6, Number 1, pp. 9–24.
9. ^ 1868 On the hypotheses which lie at the foundation of geometry, translated by W.K.Clifford, Nature 8 1873 183 – reprinted in Clifford's Collected Mathematical Papers, London 1882 (MacMillan); New York 1968 (Chelsea) http://www.emis.de/classics/Riemann/. Also in Ewald, William B., ed., 1996 "From Kant to Hilbert: A Source Book in the Foundations of Mathematics", 2 vols. Oxford Uni. Press: 652–61.
10. ^ Christoffel, E.B. (1869). "Ueber die Transformation der homogenen Differentialausdrücke zweiten Grades". Journal für die Reine und Angewandte Mathematik. 70.
11. S2CID 120009332
    .
12. ^ ^{a b} Dieudonné, J., 2009. A history of algebraic and differential topology, 1900-1960. Springer Science & Business Media.
13. ^ ^{a b} Fré, P.G., 2018. A Conceptual History of Space and Symmetry. Springer, Cham.
14. ^ The area preserving condition (or the twisting condition) cannot be removed. If one tries to extend such a theorem to higher dimensions, one would probably guess that a volume preserving map of a certain type must have fixed points. This is false in dimensions greater than 3.
15. ^ Hestenes, David (2011). "The Shape of Differential Geometry in Geometric Calculus" (PDF). In Dorst, L.; Lasenby, J. (eds.). Guide to Geometric Algebra in Practice. Springer Verlag. pp. 393–410. Archived (PDF) from the original on 2014-08-14. There is also a pdf^{[permanent dead link]} available of a scientific talk on the subject
16. ISBN 978-0-521-65116-5
    .
17. S2CID 12265584
    .
18. ISBN 978-1-4419-1968-7
    .
19. ^ Micheli, Mario (May 2008). The Differential Geometry of Landmark Shape Manifolds: Metrics, Geodesics, and Curvature (PDF) (Ph.D.). Archived from the original (PDF) on June 4, 2011.
20. ^ Joshi, Anand A. (August 2008). Geometric Methods for Image Processing and Signal Analysis (PDF) (Ph.D.). Archived (PDF) from the original on 2011-07-20.
21. doi:10.1109/TIT.2003.817466. Archived from the original
    (PDF) on 2008-10-02.

Further reading

• Ethan D. Bloch (27 June 2011). A First Course in Geometric Topology and Differential Geometry. Boston: Springer Science & Business Media.
  OCLC 811474509
  .
• Burke, William L. (1997). Applied differential geometry. Cambridge University Press.
  OCLC 53249854
  .
• OCLC 1529515
  .
• OCLC 51855212
  .
• Elsa Abbena; Simon Salamon; Alfred Gray (2017). Modern Differential Geometry of Curves and Surfaces with Mathematica (3rd ed.). Boca Raton: Chapman and Hall/CRC.
  OCLC 1048919510
  .
• Kreyszig, Erwin (1991). Differential Geometry. New York: Dover Publications.
  OCLC 23384584
  .
• Kühnel, Wolfgang (2002). Differential Geometry: Curves – Surfaces – Manifolds (2nd ed.). Providence, R.I.: American Mathematical Society.
  OCLC 61500086
  .
• McCleary, John (1994). Geometry from a differentiable viewpoint. Cambridge University Press.
  OCLC 915912917
  .
• OCLC 179192286
  .
• ter Haar Romeny, Bart M. (2003). Front-end vision and multi-scale image analysis : multi-scale computer vision theory and applications, written in Mathematica. Dordrecht: Kluwer Academic.
  OCLC 52806205
  .

External links

• "Differential geometry", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• B. Conrad. Differential Geometry handouts, Stanford University
• Michael Murray's online differential geometry course, 1996 Archived 2013-08-01 at the Wayback Machine
• A Modern Course on Curves and Surfaces, Richard S Palais, 2003 Archived 2019-04-09 at the Wayback Machine
• Richard Palais's 3DXM Surfaces Gallery Archived 2019-04-09 at the Wayback Machine
• Balázs Csikós's Notes on Differential Geometry Archived 2009-06-05 at the Wayback Machine
• N. J. Hicks, Notes on Differential Geometry, Van Nostrand.
• MIT OpenCourseWare: Differential Geometry, Fall 2008