mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

and theoretical physics, the induced metric is the

metric tensor In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...

defined on a submanifold that is induced from the metric tensor on a

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...

into which the submanifold is embedded, through the pullback. It may be determined using the following formula (using the

Einstein summation convention In mathematics, especially the usage of linear algebra in Mathematical physics, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of i ...

), which is the component form of the pullback operation: :

g_ = \partial_a X^\mu \partial_b X^\nu  g_\

Here

a

b

describe the indices of coordinates

\xi^a

of the submanifold while the functions

X^\mu(\xi^a)

encode the embedding into the higher-dimensional manifold whose tangent indices are denoted

\mu

\nu

Example – curve on a torus

Let :

\Pi\colon \mathcal \to \mathbb^3,\ \tau \mapsto \begin\beginx^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 \end

be a map from the domain of the curve

\mathcal

with parameter

\tau

into the Euclidean manifold

\mathbb^3

. Here

a,b,m,n\in\mathbb

are constants. Then there is a metric given on

\mathbb^3

as :

g=\sum\limits_g_\mathrmx^\mu\otimes \mathrmx^\nu\quad\text\quad
g_ = \begin1 & 0 & 0\\0 & 1 & 0\\0 & 0 & 1\end

. and we compute :

g_=\sum\limits_\frac\frac\underbrace_ = \sum\limits_\mu\left(\frac\right)^2=m^2 a^2+2m^2ab\cos(n\cdot \tau)+m^2b^2\cos^2(n\cdot \tau)+b^2n^2

Therefore

g_\mathcal=(m^2 a^2+2m^2ab\cos(n\cdot \tau)+m^2b^2\cos^2(n\cdot \tau)+b^2n^2) \, \mathrm\tau\otimes \mathrm\tau

References

{{Reflist Differential geometry

Example – curve on a torus

See also

References