Beltrami's theorem

Last updated January 30, 2024

In the mathematical field of differential geometry, any (pseudo-)Riemannian metric determines a certain class of paths known as geodesics. Beltrami's theorem, named for Italian mathematician Eugenio Beltrami, is a result on the inverse problem of determining a (pseudo-)Riemannian metric from its geodesics.

It is nontrivial to see that, on any Riemannian manifold of constant curvature, there are smooth coordinates relative to which all nonconstant geodesics appear as straight lines. In the negative curvature case of hyperbolic geometry, this is justified by the Beltrami–Klein model. In the positive curvature case of spherical geometry, it is justified by the gnomonic projection. In the language of projective differential geometry, these charts show that any Riemannian manifold of constant curvature is locally projectively flat. More generally, any pseudo-Riemannian manifold of constant curvature is locally projectively flat.^[1]

Beltrami's theorem asserts the converse: any connected pseudo-Riemannian manifold which is locally projectively flat must have constant curvature.^[2] With the use of tensor calculus, the proof is straightforward. Hermann Weyl described Beltrami's original proof (done in the two-dimensional Riemannian case) as being much more complicated.^[3] Relative to a projectively flat chart, there are functions $ρ i$ such that the Christoffel symbols take the form

\Gamma _{ij}^{k}=\rho _{i}\delta _{j}^{k}+\rho _{j}\delta _{i}^{k}.

Direct calculation then shows that the Riemann curvature tensor is given by

R_{ijkl}=(\partial _{i}\rho _{j}-\partial _{j}\rho _{i})g_{kl}+g_{jl}(\partial _{i}\rho _{k}-\rho _{i}\rho _{k})-g_{il}(\partial _{j}\rho _{k}-\rho _{j}\rho _{k}).

The curvature symmetry $R ijkl + R jikl = 0$ implies that $\partial i ρ j = \partial j ρ i$ . The other curvature symmetry $R ijkl = R klij$ , traced over $i$ and $l$ , then says that

\partial _{j}\rho _{k}-\rho _{j}\rho _{k}=g_{jk}{\frac {g^{il}(\partial _{i}\rho _{l}-\rho _{i}\rho _{l})}{n}}

where $n$ is the dimension of the manifold. It is direct to verify that the left-hand side is a (locally defined) Codazzi tensor, using only the given form of the Christoffel symbols. It follows from Schur's lemma that $g il (\partial i ρ l - ρ i ρ l)$ is constant. Substituting the above identity into the Riemann tensor as given above, it follows that the chart domain has constant sectional curvature $−.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num,.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0 0.1em}.mw-parser-output .sfrac .den{border-top:1px solid}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}1/ngil(∂iρl − ρiρl)$ . By connectedness of the manifold, this local constancy implies global constancy.

Beltrami's theorem may be phrased in the language of geodesic maps: if given a geodesic map between pseudo-Riemannian manifolds, one manifold has constant curvature if and only if the other does.

Related Research Articles

In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor is the most common way used to express the curvature of Riemannian manifolds. It assigns a tensor to each point of a Riemannian manifold. It is a local invariant of Riemannian metrics which measures the failure of the second covariant derivatives to commute. A Riemannian manifold has zero curvature if and only if it is flat, i.e. locally isometric to the Euclidean space. The curvature tensor can also be defined for any pseudo-Riemannian manifold, or indeed any manifold equipped with an affine connection.

In Riemannian or pseudo-Riemannian geometry, the Levi-Civita connection is the unique affine connection on the tangent bundle of a manifold that preserves the (pseudo-)Riemannian metric and is torsion-free.

In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measure of the degree to which the geometry of a given metric tensor differs locally from that of ordinary Euclidean space or pseudo-Euclidean space.

In the mathematical field of Riemannian geometry, the scalar curvature is a measure of the curvature of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the geometry of the metric near that point. It is defined by a complicated explicit formula in terms of partial derivatives of the metric components, although it is also characterized by the volume of infinitesimally small geodesic balls. In the context of the differential geometry of surfaces, the scalar curvature is twice the Gaussian curvature, and completely characterizes the curvature of a surface. In higher dimensions, however, the scalar curvature only represents one particular part of the Riemann curvature tensor.

In mathematics, the uniformization theorem states that every simply connected Riemann surface is conformally equivalent to one of three Riemann surfaces: the open unit disk, the complex plane, or the Riemann sphere. The theorem is a generalization of the Riemann mapping theorem from simply connected open subsets of the plane to arbitrary simply connected Riemann surfaces.

In the mathematical field of differential geometry, Ricci-flatness is a condition on the curvature of a (pseudo-)Riemannian manifold. Ricci-flat manifolds are a special kind of Einstein manifold. In theoretical physics, Ricci-flat Lorentzian manifolds are of fundamental interest, as they are the solutions of Einstein's field equations in vacuum with vanishing cosmological constant.

In mathematics, specifically differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension greater than 2 is too complicated to be described by a single number at a given point. Riemann introduced an abstract and rigorous way to define curvature for these manifolds, now known as the Riemann curvature tensor. Similar notions have found applications everywhere in differential geometry of surfaces and other objects. The curvature of a pseudo-Riemannian manifold can be expressed in the same way with only slight modifications.

In the mathematical field of differential geometry, there are various splitting theorems on when a pseudo-Riemannian manifold can be given as a metric product. The best-known is the Cheeger–Gromoll splitting theorem for Riemannian manifolds, although there has also been research into splitting of Lorentzian manifolds.

In differential geometry, the Weyl curvature tensor, named after Hermann Weyl, is a measure of the curvature of spacetime or, more generally, a pseudo-Riemannian manifold. Like the Riemann curvature tensor, the Weyl tensor expresses the tidal force that a body feels when moving along a geodesic. The Weyl tensor differs from the Riemann curvature tensor in that it does not convey information on how the volume of the body changes, but rather only how the shape of the body is distorted by the tidal force. The Ricci curvature, or trace component of the Riemann tensor contains precisely the information about how volumes change in the presence of tidal forces, so the Weyl tensor is the traceless component of the Riemann tensor. This tensor has the same symmetries as the Riemann tensor, but satisfies the extra condition that it is trace-free: metric contraction on any pair of indices yields zero. It is obtained from the Riemann tensor by subtracting a tensor that is a linear expression in the Ricci tensor.

In differential geometry, the Cotton tensor on a (pseudo)-Riemannian manifold of dimension n is a third-order tensor concomitant of the metric. The vanishing of the Cotton tensor for n = 3 is necessary and sufficient condition for the manifold to be locally conformally flat. By contrast, in dimensions n ≥ 4, the vanishing of the Cotton tensor is necessary but not sufficient for the metric to be conformally flat; instead, the corresponding necessary and sufficient condition in these higher dimensions is the vanishing of the Weyl tensor, while the Cotton tensor just becomes a constant times the divergence of the Weyl tensor. For n < 3 the Cotton tensor is identically zero. The concept is named after Émile Cotton.

When studying and formulating Albert Einstein's theory of general relativity, various mathematical structures and techniques are utilized. The main tools used in this geometrical theory of gravitation are tensor fields defined on a Lorentzian manifold representing spacetime. This article is a general description of the mathematics of general relativity.

In the mathematical field of differential geometry, a smooth map between Riemannian manifolds is called harmonic if its coordinate representatives satisfy a certain nonlinear partial differential equation. This partial differential equation for a mapping also arises as the Euler-Lagrange equation of a functional called the Dirichlet energy. As such, the theory of harmonic maps contains both the theory of unit-speed geodesics in Riemannian geometry and the theory of harmonic functions.

In the mathematical field of differential geometry, the existence of isothermal coordinates for a (pseudo-)Riemannian metric is often of interest. In the case of a metric on a two-dimensional space, the existence of isothermal coordinates is unconditional. For higher-dimensional spaces, the Weyl–Schouten theorem characterizes the existence of isothermal coordinates by certain equations to be satisfied by the Riemann curvature tensor of the metric.

In Riemannian geometry, a branch of mathematics, harmonic coordinates are a certain kind of coordinate chart on a smooth manifold, determined by a Riemannian metric on the manifold. They are useful in many problems of geometric analysis due to their regularity properties.

In differential geometry, a Kähler–Einstein metric on a complex manifold is a Riemannian metric that is both a Kähler metric and an Einstein metric. A manifold is said to be Kähler–Einstein if it admits a Kähler–Einstein metric. The most important special case of these are the Calabi–Yau manifolds, which are Kähler and Ricci-flat.

In mathematics, specifically in differential geometry, isothermal coordinates on a Riemannian manifold are local coordinates where the metric is conformal to the Euclidean metric. This means that in isothermal coordinates, the Riemannian metric locally has the form

In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensively studied from various perspectives: extrinsically, relating to their embedding in Euclidean space and intrinsically, reflecting their properties determined solely by the distance within the surface as measured along curves on the surface. One of the fundamental concepts investigated is the Gaussian curvature, first studied in depth by Carl Friedrich Gauss, who showed that curvature was an intrinsic property of a surface, independent of its isometric embedding in Euclidean space.

In differential geometry, Cohn-Vossen's inequality, named after Stefan Cohn-Vossen, relates the integral of Gaussian curvature of a non-compact surface to the Euler characteristic. It is akin to the Gauss–Bonnet theorem for a compact surface.

In the mathematical field of differential geometry, a biharmonic map is a map between Riemannian or pseudo-Riemannian manifolds which satisfies a certain fourth-order partial differential equation. A biharmonic submanifold refers to an embedding or immersion into a Riemannian or pseudo-Riemannian manifold which is a biharmonic map when the domain is equipped with its induced metric. The problem of understanding biharmonic maps was posed by James Eells and Luc Lemaire in 1983. The study of harmonic maps, of which the study of biharmonic maps is an outgrowth, had been an active field of study for the previous twenty years. A simple case of biharmonic maps is given by biharmonic functions.

References

↑ Schouten 1954, p. 292.
↑ do Carmo 2016, p. 301; Eisenhart 1926, Section 40; Schouten 1954, Section VI.2; Struik 1961, Section 5-3.
↑ Beltrami 1868; Weyl 1921, Footnote on p. 110.

Sources.

Beltrami, Eugenio (1868). "Teoria fondamentale degli spazii di curvature costante". Annali di Matematica Pura ed Applicata . Serie II. 2 (1): 232–255. doi:10.1007/BF02419615. JFM 01.0208.03. S2CID 120773141.
do Carmo, Manfredo P. (2016). Differential geometry of curves & surfaces (Revised & updated second edition of 1976 original ed.). Mineola, NY: Dover Publications, Inc. ISBN 978-0-486-80699-0. MR 3837152. Zbl 1352.53002.
Eisenhart, Luther Pfahler (1926). Riemannian geometry. Reprinted in 1997. Princeton: Princeton University Press. doi:10.1515/9781400884216. ISBN 0-691-02353-0. JFM 52.0721.01.
Schouten, J. A. (1954). Ricci-calculus. An introduction to tensor analysis and its geometrical applications. Die Grundlehren der mathematischen Wissenschaften. Vol. 10 (Second edition of 1923 original ed.). Berlin–Göttingen–Heidelberg: Springer-Verlag. doi:10.1007/978-3-662-12927-2. ISBN 978-3-540-01805-6. MR 0066025. Zbl 0057.37803.
Struik, Dirk J. (1961). Lectures on classical differential geometry. Reprinted in 1988. (Second edition of 1950 original ed.). London: Addison-Wesley Publishing Co. ISBN 0-486-65609-8. MR 0939369. Zbl 0105.14707.
Weyl, H. (1921). "Zur Infinitesimalgeometrie: Einordnung der projektiven und der konformen Auffassung". Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse: 99–112. JFM 48.0844.04.

External links

Weisstein, Eric W. "Beltrami's theorem". MathWorld .

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[FOOTNOTESchouten1954292-1] Schouten 1954, p. 292.

[FOOTNOTEdo_Carmo2016301Eisenhart1926Section_40Schouten1954Section_VI.2Struik1961Section_5-3-2] Carmo 2016, p. 301; Eisenhart 1926, Section 40; Schouten 1954, Section VI.2; Struik 1961, Section 5-3.

[FOOTNOTEBeltrami1868Weyl1921Footnote_on_p._110-3] Beltrami 1868; Weyl 1921, Footnote on p. 110.

[1]

[2]

[3]