Differential geometry

In mathematics, the differential geometry is the study of geometry using the tools of mathematical analysis and multilineal algebra. The study objects of this field are the differentiable varieties, which generalize the notion of surface in the euclid space, as well as the differentiable applications between them. The varieties do not have to have a natural geometric interpretation, nor do they have to be immersed in a surrounding space: for example, the general linear group GL(n,R){displaystyle GL(n,mathbb {R}}}} has differentiable variety structure, but not an intuitive geometric interpretation.

Whereas differential topology focuses solely on the topological properties of manifolds, differential geometry allows known results of multivariate calculus to be applied to applications between manifolds. Furthermore, it is possible to ascribe to any manifold geometric properties such as distances and angles if it is endowed with a Riemann metric; and features such as geodesics and curvature if a connection is added.

Differential geometry has important applications in physics, especially in the study of the theory of general relativity, where spacetime is described as a differentiable manifold.

Differential geometry of curves and surfaces

Differentiable varieties

A variety is a mathematical object that generalizes the notions of curves and surfaces to objects of more than two dimensions, not necessarily embedded in the Euclide space. Intuitively, a variety M{displaystyle M} is a set that is locally similar to eucliding space Rn{displaystyle mathbb {R} ^{n} dimension n{displaystyle n}for a certain positive integer n{displaystyle n} called dimension of variety.

The way to describe this relationship between both sets is through collections of functions, called letters. The collection of these letters is called atlas. A wing A{displaystyle {mathcal {A}}} for a variety M{displaystyle M} is a collection of pairs A={(Uα α ,φ φ α α ):α α 한 한 I!{displaystyle {mathcal {A}}={(U_{alpha },varphi _{alpha }):Where

• each set Uα α M{displaystyle U_{alpha }subset M} is an open environment of variety.
• the union of all open Uα α {displaystyle U_{alpha }} coating M{displaystyle M}: α α 한 한 IUα α =M{displaystyle bigcup _{alpha in I}U_{alpha }=M}.
• each function φ φ α α :Uα α → → Vα α Rn{displaystyle varphi _{alpha }:U_{alpha }to V_{alpha }subset mathbb {R} ^{n}} It's bijective.

To functions φ φ α α {displaystyle varphi _{alpha }} is called coordinate functions. For each pair of indexes α α ,β β 한 한 I{displaystyle alphabeta in I}, function

Δ Δ α α ,β β =φ φ β β φ φ α α − − 1:(Vα α Vβ β )→ → (Vα α Vβ β ){displaystyle tau _{alphabeta }=varphi _{beta }circ varphi _{alpha }^{-1}:(V_{alpha }cap V_{beta })to (V_{alpha }cap V_{beta }}}}

is well defined when the images of both cards have a non-empty intersection. These functions are called transition functions, and they are real functions of several variables, whose properties are well known. Depending on what properties these functions have, we will talk about one type of variety or another.

Based on a variety M{displaystyle M} successive levels of structure can be defined that add additional properties. In general, these depend on the properties that are conserved by the transitional functions; in other cases it is necessary to explicitly specify the additional structure:

• Topological variety structure, if a topology is defined M{displaystyle M} that is compatible with the cards (i.e., that the coordinate functions are homeomorphisms). It is often also required that M{displaystyle M} be a space of Hausdorff and satisfy the second axiom of numerability.
• Differentiable variety structure, if atlas is differential, i.e. transition functions are different. In such a case, transitional functions are said to be compatible; card compatibility is an equivalence ratio. Analytical varieties and dianalytic varieties can also be defined (over C{displaystyle mathbb {C} } and C+{displaystyle mathbb {C} ^{+}).
• a Riemannian metric, which is an internal product defined for each tangent space, and which varies gently from point to point. This structure allows you to define the notions of distance and angle in the variety.
• a connection specifies how to connect the environment from one point to the environment of another. It allows to define a type of derivation of interest in differential geometry: the covariant derivative.

Applications that can be distinguished between varieties

When two varieties have differentiable variety structure, then we can define the notion of differential application between them. Be two varieties M{displaystyle M} and N{displaystyle N}, of dimensions m{displaystyle m} and n{displaystyle n}, with differential structure regarding the atlas {(Uα α ,φ φ α α ):α α 한 한 I!{displaystyle {(U_{alpha },varphi _{alpha }): and {(Wβ β ,END END β β ):β β 한 한 J!{displaystyle {(W_{beta },psi _{beta }):beta in J}.

It is said that an application f:M→ → N{displaystyle f:Mto N} That's it. differential at a point p if for all pairs of cards (Uα α ,φ φ α α ){displaystyle (U_{alpha },varphi _{alpha })} and (Wβ β ,END END α α ){displaystyle (W_{beta },psi _{alpha })}focused on p and f(p) respectively, composition

F=END END β β f φ φ α α − − 1:Vα α → → Wβ β {displaystyle F=psi _{beta }circ fcirc varphi _{alpha} ^{-1}:V_{alpha }to W_{beta }}}}

is differentiated as multivariate function F:Vα α Rm→ → Wβ β Rn{displaystyle F:V_{alpha }subset mathbb {R} ^{m}to W_{beta }subset mathbb {R} ^{n}}. It is said that the application is differential' if it is differential at all points M{displaystyle M}. The fact that the transition functions are differential ensures that the definition does not depend on the chosen letters.

It has the following properties:

• The composition of two differentiable functions is different.
• The coordinate functions are differentiated, and therefore dimorphoses.

Tangent varieties

Additional bibliography

• Ethan D. Bloch (27 June 2011). A First Course in Geometric Topology and Differential Geometry (in English). Boston: Springer Science & Business Media. ISBN 978-0-8176-8122-7. OCLC 811474509.
• Burke, William L. (1997). Applied differential geometry (in English). Cambridge University Press. ISBN 0-521-26929-6. OCLC 53249854.
• Frankel, Theodore (2004). The geometry of physics: an introduction (in English) (2nd edition). New York: Cambridge University Press. ISBN 978-0-521-53927-2. OCLC 51855212.
• Elsa Abbena; Simon Salamon; Alfred Gray (2017). Modern Differential Geometry of Curves and Surfaces with Mathematica. (in English) (3rd edition). Boca Raton: Chapman and Hall/CRC. ISBN 978-1-351-99220-6. OCLC 1048919510.
• Kreyszig, Erwin (1991). Differential Geometry (in English). New York: Dover Publications. ISBN 978-0-486-66721-8. OCLC 23384584.
• Kühnel, Wolfgang (2002). Differential Geometry: Curves – Surfaces – Manifolds (in English) (2nd edition). Providence, R.I.: American Mathematical Society. ISBN 978-0-8218-3988-1. OCLC 61500086.
• McCleary, John (1994). Geometry from a differentiable viewpoint (in English). Cambridge University Press. ISBN 0-521-13311-4. OCLC 915912917.
• Spivak, Michael (1999). A Comprehensive Introduction to Differential Geometry (5 Volumes) (in English) (3rd edition). Publish or Perish. ISBN 0-914098-72-1. 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 (in English). Dordrecht: Kluwer Academic. ISBN 978-1-4020-1507-6. OCLC 52806205.