A bilinear (0,2) tensor field **g**: **V**×**V**→**R**
on a manifold M that is symmetric and non-degenerate. One can think of the
metric as an object **g**(_,_) that takes two vectors **u** and **v**,
and returns a real number, **g**(**u**,**v**). We may associate this
number with the scalar product or dot product of **u** and **v**, in
which case the metric gives shape to an otherwise shapeless topological space
by defining the lengths of tangent vectors and the angles between them.
The metric is an important concept in differential geometry.

The metric is bilinear in that it is linear
in both its arguments:

**g**(a **u** + b **v**, **w**) = a
**g**(**u**,**w**) + b **g**(**v**,**w**)

**g**(**u**, a **v + **b **w**) = a
**g**(**u**,**v**) + b **g**(**u**,**w**)

It is symmetric in that the result is
independent on the order in which it operates on two vectors: **g**(**u**,**v**)
= **g**(**v**,**u**). We impose this requirement so that **u**.**v**
= **v**.**u** and so that **v**.**v** = ||v||^{2}. is well
defined.

It is also non-degenerate in that
**g**(**u**,**v**) = 0, ∀**u** ∈**V** ↔ **v**
= 0, i.e. if
**g**(**u**,**v**) = 0 for all **u** then **v** must be zero, and so with the converse. unperson kindly points out that **g**(**u**,**v**) = 0 implies that either **u** or **v** or both are zero, or that they are perpendicular to one another.

As an example, in Euclidean **R**^{3}
we may write

**g**( , ) = **d**x^{1}(
)**d**x^{1}( ) + **d**x^{2}( )**d**x^{2}(
) + **d**x^{3}( )**d**x^{3}( )

where **d**x^{α}( ) are
the basis oneforms. Here it is understood that the first oneform in each
term acts on the first argument of **g**, while the second oneform acts
on the second argument. Strictly, **d**x^{1}(
)**d**x^{1}( ) is a tensor product **d**x^{1}(
)⊗**d**x^{1}( ). For instance, let **u** = ∂/∂x^{1}
+ 2 ∂/∂x^{2} + 3 ∂/∂x^{3} and **v** = 2 ∂/∂x^{1} -
∂/∂x^{2} + ∂/∂x^{3}. Then

**g**(**u**,**v**) =

**d**x^{1}(∂/∂x^{1} +
2 ∂/∂x^{2} + 3 ∂/∂x^{3})**d**x^{1}(2
∂/∂x^{1} - ∂/∂x^{2} + ∂/∂x^{3})
+

**d**x^{2}(∂/∂x^{1} + 2 ∂/∂x^{2} + 3 ∂/∂x^{3})**d**x^{2}(2
∂/∂x^{1} - ∂/∂x^{2} + ∂/∂x^{3})
+

**d**x^{3}(∂/∂x^{1} + 2 ∂/∂x^{2} + 3 ∂/∂x^{3})**d**x^{3}(2
∂/∂x^{1} - ∂/∂x^{2} + ∂/∂x^{3})

Since **d**x^{1}(∂/∂x^{1})
= 1 and **d**x^{2}(∂/∂x^{1})
= 0, etc. So
**g**(**u**,**v**) = (1)(2) + (2)(-1) + (3)(1) = 3, or **u**.**v**
= 3.

We can also employ the short-hand notation **g = ** **d**x^{1}**d**x^{1}
+** d**x^{2}**d**x^{2} + **d**x^{3}**d**x^{3}
where it is again understood that the first oneform in each term acts on the
first argument of **g**, and so on. You may sometimes see this written as ds^{2}
= dx^{2} + dy^{2} + dz^{2} which is suggestive of the pythagorean
theorem applied to the differentials dx, dy and dz, where ds is some
differential length, however this isn't very rigorous although it may serve as
a memory aid.

Another way to use the metric is to act on only one vector, leaving the other
argument unoccupied, e.g.
**g**(**u**,** **). The result is then an object that
takes another vector **v** and returns a real number, with precisely the same
properties as a oneform. Thus the metric provides a natural relationship
between vectors and oneforms and a way to associate a unique oneform to
every vector, e.g. **μ** =
**g**(**u**,** **) where **μ** is the oneform
associated with the vector **u**. For instance, consider again **u** = ∂/∂x^{1}
+ 2 ∂/∂x^{2} + 3 ∂/∂x^{3}. Then

**g**(**u**, ) =

**d**x^{1}(∂/∂x^{1} +
2 ∂/∂x^{2} + 3 ∂/∂x^{3})**d**x^{1}
+

**d**x^{2}(∂/∂x^{1} + 2 ∂/∂x^{2} + 3 ∂/∂x^{3})**d**x^{2}
+

**d**x^{3}(∂/∂x^{1} + 2 ∂/∂x^{2} + 3 ∂/∂x^{3})**d**x^{3}

So **μ** =
**g**(**u**,** **) = **d**x^{1}
+ 2 **d**x^{2}
+ 3 **d**x^{3}. Since we are dealing with Euclidean **R**^{3}
the relationship between a vector and its associated oneform is trivial, as the
components are unaltered, however the example serves its illustrative purpose.

It would then be natural to have a way to
associate a vector with any oneform such that the mapping is one-to-one
and onto. That is, so that every vector **u** maps to a unique oneform
**μ**, and for every oneform **μ** there is one and only
one vector that maps to it, namely **u**. To do this we introduce the
inverse metric **g**^{-1} such that **g**^{-1}(**g**(**u**,
), ) = **u**. We can then say associate with every oneform **μ**
a unique vector **u** = **g**^{-1}(**μ**,
). Again, in Euclidean **R**^{3} we may write

**g**^{-1}( , ) = ∂/∂x^{1}(
)∂/∂x^{1}( ) + ∂/∂x^{2}( )∂/∂x^{2}(
) + ∂/∂x^{3}( )∂/∂x^{3}( )

where **g**^{-1} takes two
oneforms as arguments and returns a real number, or a single oneform to return a
vector.

The metric becomes more useful in other coordinate systems and spaces. For
instance, consider the cylindrical parameterization of
**R**^{3} in terms of r, θ, and z:

x^{1} = r cosθ

x^{2} = r sinθ

x^{3} = z

Then **d**x^{1} = ∂x^{1}/∂r
**d**r + ∂x^{1}/∂θ **d**θ
+ ∂x^{1}/∂z **d**z
= cosθ**d**r - r sinθ **d**θ, **d**x^{2}
= sinθ**d**r + r cosθ **d**θ, and **d**x^{3}
= **d**z. Then

**g** = **d**x^{1}**d**x^{1} + **d**x^{2}**d**x^{2} + **d**x^{3}**d**x^{3
}= (cosθ**d**r - r sinθ **d**θ)(cosθ** d**r
- r sinθ **d**θ) + (sinθ**d**r + r
cosθ **d**θ)(sinθ**d**r + r
cosθ **d**θ)
+ **d**z** d**z

= cos^{2}θ** d**r** d**r - r sinθ
cosθ** d**θ
**d**r - r sinθ cosθ** d**r **d**θ + r^{2}
sin^{2}θ** d**θ **d**θ
+ sin^{2}θ** d**r** d**r + r
sinθ cosθ** d**θ
**d**r + r sinθ cosθ** d**r **d**θ + r^{2}
cos^{2}θ** d**θ **d**θ
+ **d**z** d**z

Thus **g **= **d**r** d**r
+ r^{2} **d**θ **d**θ
+ **d**z** d**z

In some coordinate systems the non-degeneracy condition breaks down in
certain places, for example, in spherical coordinates **g **
= **d**r** d**r
+ r^{2} **d**θ **d**θ
+ r^{2 }sin^{2}θ **d**φ **d**φ, the metric
becomes degenerate where sin^{2}θ = 0, or at θ = 0 or π.
In such cases additional parameterizations may be necessary to completely cover
the entire manifold.

It is useful to express the metric tensor in abstract index notation, so in
terms of basis oneforms e^{α}_{a}, where α denotes the α^{th} basis
oneform,
we may write:

g_{ab} = g_{αβ} e^{α}_{a }e^{β}_{b}

where g_{αβ} are the components of the metric tensor, and
greek indices that appear both as superscript and subscript are summed over. In
this notation, a denotes part of the tensor that acts on the corresponding
vector labeled a, while b denotes the part that acts on the corresponding vector
labeled b. We may then write the action of **g** on two vectors **u** and
**v** as g_{ab} u^{a} v^{b}. It is understood
that **u** corresponds to the first argument of **g**, while **v**
corresponds to the second argument of **g**, regardless of the order that
they may appear in the expression, e.g. g_{ab} u^{a} v^{b} =
g_{ab} v^{b} u^{a} = g_{dc} u^{d} v^{c}
. Since the metric is symmetric the distinction is not an important one, but
is useful when **g** acts on objects other than vectors. In terms of
components and the basis vectors, u^{a} = u^{α }e_{α}^{a}, and
v^{b} = v^{β }e_{β}^{b}, then

g_{ab} u^{a} v^{b} = g_{αβ} e^{α}_{a }e^{β}_{b }u^{γ }e_{γ}^{a} v^{δ }e_{δ}^{b}

Rearranging yields

g_{ab} u^{a} v^{b} = g_{αβ} _{ }u^{γ} v^{δ}^{ }e^{α}_{a }e_{γ}^{a} e_{δ}^{b} e^{β}_{b}

Since e^{α}_{a }e_{γ}^{a} = δ^{α}_{γ},
etc. where δ^{α}_{γ} is the Kronecker delta we have

g_{ab} u^{a} v^{b} = g_{αβ} _{ }u^{γ} v^{δ }δ^{α}_{γ} δ_{δ}^{β}

g_{ab} u^{a} v^{b} = g_{αβ} _{ }u^{α} v^{α}

We may also write the inverse metric **g**^{-1} in terms of the
basis vectors e_{α}^{a} as

g^{ab} = g^{αβ} e_{α}^{a}_{ }e_{β}^{b}

With both the metric and its inverse, we can find the oneform that
corresponds to any vector, and vice versa. Thus the metric can be thought of
an object that raises or lowers indices of oneforms and vectors. We can then
also use the metric and its inverse to raise and lower the indices of higher
order tensors. Consider the (1,1) tensor T^{a}_{b} = T^{α}_{β }e_{α}^{a} e^{β}_{b.} Suppose we wish to calculate
T_{a}^{b}, that is lower the first index while raising the
second. We can use g_{ab} to lower the first index, and then use g^{ab}
to raise the second index, as follows:

T_{a}^{b} = g_{ac} g^{db} T^{c}_{d}

(Components have been suppressed for simplicity.)

Although the above discussion has mostly dealt with Euclidean **R**^{3}
in Cartesian coordinates, it is general enough to be extended to other spaces
easily. Starting in **R**^{n} with the coordinate basis oneforms **d**x^{α}
in Cartesian coordinates we can express the metric as

g_{ab} = δ_{αβ} dx^{α}_{a }
dx^{β}_{b}

We can then embed a manifold of dimension m ≤ n in **R**^{n}
using the m coordinates ξ^{α}. On the manifold where x^{α}
are smooth, invertible functions of ξ^{α}, i.e. the Jacobian
Determinant is nonzero, we can write dx^{α}_{a} = ∂x^{α}/∂x^{δ}dξ^{δ}_{a}.
Thus

g_{ab} = δ_{αβ} ∂x^{α}/∂x^{δ}dξ^{δ}_{a }
∂x^{β}/∂x^{ε}dξ^{ε}_{b},
or

g_{ab} = δ_{αβ} ∂x^{α}/∂x^{δ }∂x^{β}/∂x^{ε} dξ^{δ}_{a }
dξ^{ε}_{b},

g_{ab} = g_{δε} dξ^{δ}_{a }
dξ^{ε}_{b},

Thus in the new coordinate system, g_{δε} = δ_{αβ } ∂x^{α}/∂x^{δ }∂x^{β}/∂x^{ε},
and we may write, after relabling indices,

g_{αβ} = Σ_{δ} (∂x^{δ}/∂x^{α
})(∂x^{δ}/∂x^{β})

Although this formula assumes an embedding into a higher-dimensional space,
the concept of the metric is an intrinsic property of the manifold and is an
important construct that embodies the curvature of the manifold. The
metric tensor can be used to specify the Christoffel Symbols (see Covariant
Derivative) as well as the Riemann Curvature tensor. A connection derived
from the metric tensor is known as a metric compatible connection. We can also
derive a metric compatible volume form (see volume form).

An example of a metic important in physics is the Minkowski metric in
Special relativity which describes flat spacetime:

**g** = - c **d**t **d**t + dx^{1}**d**x^{1} + **d**x^{2}**d**x^{2} + **d**x^{3}**d**x^{3}

where t is the temporal (time) coordinate and x^{α} re
spatial coordinates. In this spacetime non-zero intervals
can have positive (timelike), negative (spacelike) and zero (null or lightlike)
magnitudes. There is also the Schwarzschild metric in General Relativity
which describes spacetimes outside of static spherically symmetric
distributions of matter, for instance a non-rotating neutral black hole.
In spherical coordinates (with coordinates in which c and G = 0):

**g **= - (1 - 2M/r) **d**t **d**t +
(1-2M/r)^{-1} **d**r** d**r
+ r^{2} **d**θ **d**θ
+ r^{2}
sin^{2}θ **d**φ **d**φ

Here the coordinate system breaks down at r = 2M which is known as the
Schwarzschild Radius for the object. In other words, if the entire mass of the
object where compressed to within this radius, becoming a Schwarzschild black
hole then the metric would break down at this radius. Although relativistic
effects become severe as one approaches this radius, spacetime itself does not
actually break down here, which was originally thought to be the case. Instead,
once inside the event horizon, an observer's future light cone lies
entirely within this radius and the observer becomes trapped. We may
reparameterize the space, for example in terms of Kruskal-Szekeres coordinates, in which the
metric still remains valid at this distance. However, the singularity at r = 0
remains and is a real singularity.

One may also speak of the signature of a metric^{1}. When the metric is diagonalized, the signature is the number of positive elements minus the number of negative elements. For instance, the Minkowski metric has signature 3-2 = 2. A metric in which all elements are positive is known as a positive definite metric or Riemannian metric. General Relativity deals with metrics with one negative element and are known as Lorentzian metrics. Metrics such as these are known as indefinite where some elements are positive and some are negative. One can also denote the signature as (p,q) where p is the number of positive elements and q is the number of negative elements. More explicitly, one may write the metric in the form, for example, -+++ for a Lorentzian metric.

- krimson Pointed out that I should mention this distinction.