Differential geometry: Difference between revisions

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.

History of development

This section possibly contains synthesis of material which does not verifiably mention or relate to the main topic. Relevant discussion may be found on the talk page. (February 2017) (Learn how and when to remove this message)

Differential geometry arose and developed as a result of and in connection to the mathematical analysis of curves and surfaces.^[1] Mathematical analysis of curves and surfaces had been developed to answer some of the nagging and unanswered questions that appeared in calculus, like the reasons for relationships between complex shapes and curves, series and analytic functions. These unanswered questions indicated greater, hidden relationships.

The general idea of natural equations for obtaining curves from local curvature appears to have been first considered by Leonhard Euler in 1736, and many examples with fairly simple behavior were studied in the 1800's.^[2]

When curves, surfaces enclosed by curves, and points on curves were found to be quantitatively, and generally, related by mathematical forms, the formal study of the nature of curves and surfaces became a field of study in its own right, with Monge's paper in 1795, and especially, with Gauss's publication of his article, titled 'Disquisitiones Generales Circa Superficies Curvas', in Commentationes Societatis Regiae Scientiarum Gottingesis Recentiores in 1827.^[3]

Initially applied to the Euclidean space, further explorations led to non-Euclidean space, and metric and topological spaces.

Branches

Riemannian geometry

Riemannian geometry studies

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

${\displaystyle M}$

${\displaystyle J:TM\rightarrow TM}$, 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

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

.

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.

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. (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 extrinsic

From the beginning and through the middle of the 18th century, differential geometry was studied from the extrinsic point of view:

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 it?). 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.^[5]

Applications

Part of a series on
Spacetime
Special relativity General relativity
Spacetime concepts Spacetime manifold Equivalence principle Lorentz transformations Minkowski space
General relativity Introduction to general relativity Mathematics of general relativity Einstein field equations
Classical gravity Introduction to gravitation Newton's law of universal gravitation
Relevant mathematics Four-vector Derivations of relativity Spacetime diagrams Differential geometry Curved spacetime Mathematics of general relativity Spacetime topology
Physics portal  Category
v t e

Below are some examples of how differential geometry is applied to other fields of science and mathematics.

• In physics, four uses will be mentioned:
  • 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.^[6]
• 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 wireless communications, Grassmannian manifolds are used for beamforming techniques in multiple antenna systems.^[11]

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

References

1. ^ http://www.encyclopediaofmath.org/index.php/Differential_geometry be referred to
2. ISBN 1-57955-008-8
   .
3. ^ 'Disquisitiones Generales Circa Superficies Curvas' (literal translation from Latin: General Investigations of Curved Surfaces), Commentationes Societatis Regiae Scientiarum Gottingesis Recentiores (literally, Recent Perspectives, Gottingen's Royal Society of Science). Volume VI, pp. 99–146. A translation of the work, by A.M.Hiltebeitel and J.C.Morehead, titled, "General Investigations of Curved Surfaces" was published 1965 by Raven Press, New York. A digitised version of the same is available at http://quod.lib.umich.edu/u/umhistmath/abr1255.0001.001 for free download, for non-commercial, personal use. In case of further information, the library could be contacted. Also, the Wikipedia article on Gauss's works in the year 1827 at could be looked at.
4. ^ 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.
5. ^ Hestenes, David (2011). "The Shape of Differential Geometry in Geometric Calculus". In Dorst, L.; Lasenby, J. (eds.). Guide to Geometric Algebra in Practice. Springer Verlag. pp. 393–410. {{cite book}}: External link in |chapterurl= (help); Unknown parameter |chapterurl= ignored (|chapter-url= suggested) (help) There is also a pdf available of a scientific talk on the subject
6. ISBN 0-521-65116-6
   .
7. doi:10.1109/ICASSP.2005.1416480. {{cite web}}: Missing or empty |url= (help
   )
8. ISBN 978-1-4419-1968-7
   .
9. ^ 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.
10. ^ Joshi, Anand A. (August 2008). Geometric Methods for Image Processing and Signal Analysis (PDF) (Ph.D.).
11. doi:10.1109/TIT.2003.817466.{{cite journal}}: CS1 maint: multiple names: authors list (link
    )

Further reading

• Bloch, Ethan D. (1996). A First Course in Geometric Topology and Differential Geometry.
• Burke, William L. (1985). Applied Differential Geometry.
• ISBN 0-13-212589-7
  . Classical geometric approach to differential geometry without tensor analysis.
• do Carmo, Manfredo (1994). Riemannian Geometry.
• Frankel, Theodore (2004). The geometry of physics: an introduction (2nd ed.).
  ISBN 0-521-53927-7
  .
• Gray, Alfred (1998). Modern Differential Geometry of Curves and Surfaces with Mathematica (2nd ed.).
• Kreyszig, Erwin (1991). Differential Geometry.
  ISBN 0-486-66721-9
  . Good classical geometric approach to differential geometry with tensor machinery.
• Kühnel, Wolfgang (2002). Differential Geometry: Curves – Surfaces – Manifolds (2nd ed.).
  ISBN 0-8218-3988-8
  .
• McCleary, John (1994). Geometry from a Differentiable Viewpoint.
• Spivak, Michael (1999). A Comprehensive Introduction to Differential Geometry (5 Volumes) (3rd ed.).
• ter Haar Romeny, Bart M. (2003). Front-End Vision and Multi-Scale Image Analysis.
  ISBN 1-4020-1507-0
  .

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
• A Modern Course on Curves and Surface, Richard S Palais, 2003
• Richard Palais's 3DXM Surfaces Gallery
• Balázs Csikós's Notes on Differential Geometry
• N. J. Hicks, Notes on Differential Geometry, Van Nostrand.
• MIT OpenCourseWare: Differential Geometry, Fall 2008