In
mathematics and
theoretical physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...
, the induced metric is the
metric tensor defined on a
submanifold
In mathematics, a submanifold of a manifold ''M'' is a subset ''S'' which itself has the structure of a manifold, and for which the inclusion map satisfies certain properties. There are different types of submanifolds depending on exactly which ...
that is induced from the metric tensor on a
manifold into which the submanifold is embedded, through the
pullback
In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward.
Precomposition
Precomposition with a function probably provides the most elementary notion of pullback: i ...
. It may be determined using the following formula (using the
Einstein summation convention), which is the component form of the pullback operation:
:
Here
,
describe the indices of coordinates
of the submanifold while the functions
encode the embedding into the higher-dimensional manifold whose tangent indices are denoted
,
.
Example – curve on a torus
Let
:
be a map from the domain of the curve
with parameter
into the Euclidean manifold
. Here
are constants.
Then there is a metric given on
as
:
.
and we compute
:
Therefore
See also
*
First fundamental form
In differential geometry, the first fundamental form is the inner product on the tangent space of a surface in three-dimensional Euclidean space which is induced canonically from the dot product of . It permits the calculation of curvature and me ...
References
{{Reflist
Differential geometry