differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...

, the notion of a

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 ...

can be extended to an arbitrary

vector bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every po ...

, and to some

principal fiber bundle In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X \times G of a space X with a group G. In the same way as with the Cartesian product, a principal bundle P is equ ...

s. This metric is often called a bundle metric, or fibre metric.

Definition

If ''M'' is a

topological manifold In topology, a branch of mathematics, a topological manifold is a topological space that locally resembles real ''n''-dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout mathe ...

and : ''E'' → ''M'' a vector bundle on ''M'', then a metric on ''E'' is a

bundle map In mathematics, a bundle map (or bundle morphism) is a morphism in the category of fiber bundles. There are two distinct, but closely related, notions of bundle map, depending on whether the fiber bundles in question have a common base space. There ...

''k'' : ''E'' ×_''M'' ''E'' → ''M'' × R from the

fiber product In category theory, a branch of mathematics, a pullback (also called a fiber product, fibre product, fibered product or Cartesian square) is the limit of a diagram consisting of two morphisms and with a common codomain. The pullback is often w ...

of ''E'' with itself to the

trivial bundle In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...

with fiber R such that the restriction of ''k'' to each fibre over ''M'' is a

nondegenerate In mathematics, a degenerate case is a limiting case of a class of objects which appears to be qualitatively different from (and usually simpler than) the rest of the class, and the term degeneracy is the condition of being a degenerate case. T ...

bilinear map In mathematics, a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments. Matrix multiplication is an example. Definition Vector spaces Let V, W ...

vector spaces In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...

.. Roughly speaking, ''k'' gives a kind of dot product (not necessarily symmetric or positive definite) on the vector space above each point of ''M'', and these products vary smoothly over ''M''.

Properties

Every vector bundle with paracompact base space can be equipped with a bundle metric. For a vector bundle of rank ''n'', this follows from the bundle charts

\phi:\pi^(U)\to U\times\mathbb^n

: the bundle metric can be taken as the pullback of the

inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...

of a metric on

\mathbb^n

; for example, the orthonormal charts of Euclidean space. The structure group of such a metric is the

orthogonal group In mathematics, the orthogonal group in dimension , denoted , is the Group (mathematics), group of isometry, distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by ...

''O''(''n'').

Example: Riemann metric

If ''M'' is a

Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...

, and ''E'' is its

tangent bundle In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...

T''M'', then the

Riemannian metric In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ''T ...

gives a bundle metric, and vice versa.

Example: on vertical bundles

If the bundle :''P'' → ''M'' is a

with group ''G'', and ''G'' is a

compact Lie group In mathematics, a compact (topological) group is a topological group whose topology realizes it as a compact topological space (when an element of the group is operated on, the result is also within the group). Compact groups are a natural gene ...

, then there exists an Ad(''G'')-invariant inner product ''k'' on the fibers, taken from the inner product on the corresponding

compact Lie algebra In the mathematical field of Lie theory, there are two definitions of a compact Lie algebra. Extrinsically and topologically, a compact Lie algebra is the Lie algebra of a compact Lie group; this definition includes tori. Intrinsically and algeb ...

. More precisely, there is a

''k'' defined on the

vertical bundle Vertical is a geometric term of location which may refer to: * Vertical direction, the direction aligned with the direction of the force of gravity, up or down * Vertical (angles), a pair of angles opposite each other, formed by two intersecting s ...

E = V''P'' such that ''k'' is invariant under left-multiplication: :

k(L_X, L_Y)=k(X,Y)

for vertical vectors ''X'', ''Y'' and ''L''_''g'' is left-multiplication by ''g'' along the fiber, and ''L''_''g*'' is the

pushforward The notion of pushforward in mathematics is "dual" to the notion of pullback, and can mean a number of different but closely related things. * Pushforward (differential), the differential of a smooth map between manifolds, and the "pushforward" op ...

. That is, ''E'' is the vector bundle that consists of the vertical subspace of the tangent of the principal bundle. More generally, whenever one has a compact group with

Haar measure In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups. This measure was introduced by Alfréd Haar in 1933, though ...

μ, and an arbitrary

''h(X,Y)'' defined at the tangent space of some point in ''G'', one can define an invariant metric simply by averaging over the entire group, i.e. by defining :

k(X,Y)=\int_G h(L_ X, L_ Y) d\mu_g

as the average. The above notion can be extended to the

associated bundle In mathematics, the theory of fiber bundles with a structure group G (a topological group) allows an operation of creating an associated bundle, in which the typical fiber of a bundle changes from F_1 to F_2, which are both topological spaces wit ...

P\times_G V

where ''V'' is a vector space transforming covariantly under some representation of ''G''.

In relation to Kaluza–Klein theory

If the base space ''M'' is also a

metric space In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...

, with metric ''g'', and the principal bundle is endowed with a

connection form In mathematics, and specifically differential geometry, a connection form is a manner of organizing the data of a connection using the language of moving frames and differential forms. Historically, connection forms were introduced by Élie Carta ...

ω, then ^*g+kω is a metric defined on the entire

''E'' = T''P''.David Bleecker,
Gauge Theory and Variational Principles
(1982) D. Reidel Publishing ''(See chapter 9'') More precisely, one writes ^*g(''X'',''Y'') = ''g''(_*''X'', _*''Y'') where _* is the

of the projection , and ''g'' is the

on the base space ''M''. The expression ''kω'' should be understood as (''kω'')(''X'',''Y'') = ''k''(''ω''(''X''),''ω''(''Y'')), with ''k'' the metric tensor on each fiber. Here, ''X'' and ''Y'' are elements of the

tangent space In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...

T''P''. Observe that the lift ^*g vanishes on the

vertical subspace In mathematics, the vertical bundle and the horizontal bundle are vector bundles associated to a smooth fiber bundle. More precisely, given a smooth fiber bundle \pi\colon E\to B, the vertical bundle VE and horizontal bundle HE are subbundles of ...

T''V'' (since _* vanishes on vertical vectors), while kω vanishes on the horizontal subspace T''H'' (since the horizontal subspace is defined as that part of the tangent space T''P'' on which the connection ω vanishes). Since the total tangent space of the bundle is a direct sum of the vertical and horizontal subspaces (that is, T''P'' = T''V'' ⊕ T''H''), this metric is well-defined on the entire bundle. This bundle metric underpins the generalized form of Kaluza–Klein theory due to several interesting properties that it possesses. The

scalar curvature In the mathematical field of Riemannian geometry, the scalar curvature (or the Ricci scalar) 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 geometr ...

derived from this metric is constant on each fiber, this follows from the Ad(''G'') invariance of the fiber metric ''k''. The scalar curvature on the bundle can be decomposed into three distinct pieces: :''R''_''E'' = ''R''_M(''g'') + ''L''(''g'', ω) + ''R''_''G''(''k'') where ''R''_''E'' is the scalar curvature on the bundle as a whole (obtained from the metric ^*g+kω above), and ''R''_M(''g'') is the scalar curvature on the base manifold ''M'' (the

Lagrangian density Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...

of the

Einstein–Hilbert action The Einstein–Hilbert action (also referred to as Hilbert action) in general relativity is the action that yields the Einstein field equations through the stationary-action principle. With the metric signature, the gravitational part of the ac ...

), and ''L''(''g'', ω) is the Lagrangian density for the Yang–Mills action, and ''R''_''G''(''k'') is the scalar curvature on each fibre (obtained from the fiber metric ''k'', and constant, due to the Ad(''G'')-invariance of the metric ''k''). The arguments denote that ''R''_M(''g'') only depends on the metric ''g'' on the base manifold, but not ω or ''k'', and likewise, that ''R''_''G''(''k'') only depends on ''k'', and not on ''g'' or ω, and so-on.

References

{{reflist Differential geometry