Induced metric

Last updated April 11, 2021

In mathematics and theoretical physics, the induced metric is the metric tensor defined on a submanifold that is induced from the metric tensor on a manifold into which the submanifold is embedded, through the pullback.^[1] It may be determined using the following formula (using the Einstein summation convention), which is the component form of the pullback operation:^[2]

Example – curve on a torus

Let

\Pi \colon {\mathcal {C}}\to \mathbb {R} ^{3},\ \tau \mapsto {\begin{cases}{\begin{aligned}x^{1}&=(a+b\cos(n\cdot \tau ))\cos(m\cdot \tau )\\x^{2}&=(a+b\cos(n\cdot \tau ))\sin(m\cdot \tau )\\x^{3}&=b\sin(n\cdot \tau ).\end{aligned}}\end{cases}}

be a map from the domain of the curve ${\mathcal {C}}$ with parameter $\tau$ into the Euclidean manifold $\mathbb {R} ^{3}$ . Here $a,b,m,n\in \mathbb {R}$ are constants.

Then there is a metric given on $\mathbb {R} ^{3}$ as

g=\sum \limits _{\mu ,\nu }g_{\mu \nu }\mathrm {d} x^{\mu }\otimes \mathrm {d} x^{\nu }\quad {\text{with}}\quad g_{\mu \nu }={\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}}

.

and we compute

g_{\tau \tau }=\sum \limits _{\mu ,\nu }{\frac {\partial x^{\mu }}{\partial \tau }}{\frac {\partial x^{\nu }}{\partial \tau }}\underbrace {g_{\mu \nu }} _{\delta _{\mu \nu }}=\sum \limits _{\mu }\left({\frac {\partial x^{\mu }}{\partial \tau }}\right)^{2}=m^{2}a^{2}+2m^{2}ab\cos(n\cdot \tau )+m^{2}b^{2}\cos ^{2}(n\cdot \tau )+b^{2}n^{2}

Therefore $g_{\mathcal {C}}=(m^{2}a^{2}+2m^{2}ab\cos(n\cdot \tau )+m^{2}b^{2}\cos ^{2}(n\cdot \tau )+b^{2}n^{2})\,\mathrm {d} \tau \otimes \mathrm {d} \tau$

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, that measures the extent to which the metric tensor is not locally isometric to that of Euclidean space. The curvature tensor can also be defined for any pseudo-Riemannian manifold, or indeed any manifold equipped with an affine connection.

In mathematics and physics, n-dimensional anti-de Sitter space (AdS_n) is a maximally symmetric Lorentzian manifold with constant negative scalar curvature. Anti-de Sitter space and de Sitter space are named after Willem de Sitter (1872–1934), professor of astronomy at Leiden University and director of the Leiden Observatory. Willem de Sitter and Albert Einstein worked together closely in Leiden in the 1920s on the spacetime structure of the universe.

Bosonic string theory is the original version of string theory, developed in the late 1960s. It is so called because it only contains bosons in the spectrum.

In physics, the Hamilton–Jacobi equation, named after William Rowan Hamilton and Carl Gustav Jacob Jacobi, is an alternative formulation of classical mechanics, equivalent to other formulations such as Newton's laws of motion, Lagrangian mechanics and Hamiltonian mechanics. The Hamilton–Jacobi equation is particularly useful in identifying conserved quantities for mechanical systems, which may be possible even when the mechanical problem itself cannot be solved completely.

The Nambu–Goto action is the simplest invariant action in bosonic string theory, and is also used in other theories that investigate string-like objects. It is the starting point of the analysis of zero-thickness string behavior, using the principles of Lagrangian mechanics. Just as the action for a free point particle is proportional to its proper time — i.e., the "length" of its world-line — a relativistic string's action is proportional to the area of the sheet which the string traces as it travels through spacetime.

In mathematics, a Killing vector field, named after Wilhelm Killing, is a vector field on a Riemannian manifold that preserves the metric. Killing fields are the infinitesimal generators of isometries; that is, flows generated by Killing fields are continuous isometries of the manifold. More simply, the flow generates a symmetry, in the sense that moving each point on an object the same distance in the direction of the Killing vector will not distort distances on the object.

In physics, the Polyakov action is an action of the two-dimensional conformal field theory describing the worldsheet of a string in string theory. It was introduced by Stanley Deser and Bruno Zumino and independently by L. Brink, P. Di Vecchia and P. S. Howe, and has become associated with Alexander Polyakov after he made use of it in quantizing the string. The action reads

In general relativity, the Gibbons–Hawking–York boundary term is a term that needs to be added to the Einstein–Hilbert action when the underlying spacetime manifold has a boundary.

In physics, a sigma model is a field theory that describes the field as a point particle confined to move on a fixed manifold. This manifold can be taken to be any Riemannian manifold, although it is most commonly taken to be either a Lie group or a symmetric space. The model may or may not be quantized. An example of the non-quantized version is the Skyrme model; it cannot be quantized due to non-linearities of power greater than 4. In general, sigma models admit (classical) topological soliton solutions, for example, the Skyrmion for the Skyrme model. When the sigma field is coupled to a gauge field, the resulting model is described by Ginzburg–Landau theory. This article is primarily devoted to the classical field theory of the sigma model; the corresponding quantized theory is presented in the article titled "non-linear sigma model".

A theoretical motivation for general relativity, including the motivation for the geodesic equation and the Einstein field equation, can be obtained from special relativity by examining the dynamics of particles in circular orbits about the earth. A key advantage in examining circular orbits is that it is possible to know the solution of the Einstein Field Equation a priori. This provides a means to inform and verify the formalism.

In geometry, the elliptic coordinate system is a two-dimensional orthogonal coordinate system in which the coordinate lines are confocal ellipses and hyperbolae. The two foci $and are generally taken to be fixed at and, respectively, on the -axis of the Cartesian coordinate system.$

Toroidal coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional bipolar coordinate system about the axis that separates its two foci. Thus, the two foci $and in bipolar coordinates become a ring of radius in the plane of the toroidal coordinate system; the -axis is the axis of rotation. The focal ring is also known as the reference circle.$

Elliptic cylindrical coordinates are a three-dimensional orthogonal coordinate system that results from projecting the two-dimensional elliptic coordinate system in the perpendicular $-direction. Hence, the coordinate surfaces are prisms of confocal ellipses and hyperbolae. The two foci and are generally taken to be fixed at and, respectively, on the -axis of the Cartesian coordinate system.$

Prolate spheroidal coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional elliptic coordinate system about the focal axis of the ellipse, i.e., the symmetry axis on which the foci are located. Rotation about the other axis produces oblate spheroidal coordinates. Prolate spheroidal coordinates can also be considered as a limiting case of ellipsoidal coordinates in which the two smallest principal axes are equal in length.

Oblate spheroidal coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional elliptic coordinate system about the non-focal axis of the ellipse, i.e., the symmetry axis that separates the foci. Thus, the two foci are transformed into a ring of radius $in the x - y plane. Oblate spheroidal coordinates can also be considered as a limiting case of ellipsoidal coordinates in which the two largest semi-axes are equal in length.$

In 3-dimensional topology, a part of the mathematical field of geometric topology, the Casson invariant is an integer-valued invariant of oriented integral homology 3-spheres, introduced by Andrew Casson.

In differential geometry, normal coordinates at a point p in a differentiable manifold equipped with a symmetric affine connection are a local coordinate system in a neighborhood of p obtained by applying the exponential map to the tangent space at p. In a normal coordinate system, the Christoffel symbols of the connection vanish at the point p, thus often simplifying local calculations. In normal coordinates associated to the Levi-Civita connection of a Riemannian manifold, one can additionally arrange that the metric tensor is the Kronecker delta at the point p, and that the first partial derivatives of the metric at p vanish.

The tetrad formalism is an approach to general relativity that generalizes the choice of basis for the tangent bundle from a coordinate basis to the less restrictive choice of a local basis, i.e. a locally defined set of four linearly independent vector fields called a tetrad or vierbein. It is a special case of the more general idea of a vielbein formalism, which is set in Riemannian geometry. This article as currently written makes frequent mention of general relativity; however, almost everything it says is equally applicable to Riemannian manifolds in general, and even to spin manifolds. Most statements hold simply by substituting arbitrary $for . In German, "vier" translates to "four", and "viel" to "many".$

In mathematics, a non-autonomous system of ordinary differential equations is defined to be a dynamic equation on a smooth fiber bundle $over . For instance, this is the case of non-relativistic non-autonomous mechanics, but not relativistic mechanics. To describe relativistic mechanics, one should consider a system of ordinary differential equations on a smooth manifold whose fibration over is not fixed. Such a system admits transformations of a coordinate on depending on other coordinates on . Therefore, it is called the relativistic system . In particular, Special Relativity on the Minkowski space is of this type.$

Lagrangian field theory is a formalism in classical field theory. It is the field-theoretic analogue of Lagrangian mechanics. Lagrangian mechanics is used to analyze the motion of a system of discrete particles each with a finite number of degrees of freedom. Lagrangian field theory applies to continua and fields, which have an infinite number of degrees of freedom.

References

↑ Lee, John M. (2006-04-06). Riemannian Manifolds: An Introduction to Curvature. Graduate Texts in Mathematics. Springer Science & Business Media. pp. 25–27. ISBN 978-0-387-22726-9. OCLC 704424444.
↑ Poisson, Eric (2004). A Relativist's Toolkit. Cambridge University Press. p. 62. ISBN 978-0-521-83091-1.

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.

[1] Lee, John M. (2006-04-06). Riemannian Manifolds: An Introduction to Curvature. Graduate Texts in Mathematics. Springer Science & Business Media. pp. 25–27. ISBN 978-0-387-22726-9. OCLC 704424444.

[2] Poisson, Eric (2004). A Relativist's Toolkit. Cambridge University Press. p. 62. ISBN 978-0-521-83091-1.

[1]

[2]

Induced metric

Contents

Example – curve on a torus

See also

Related Research Articles

References