In differential geometry
, a Riemannian manifold or Riemannian space is a real
, smooth manifold
''M'' equipped with a positive-definite inner product
on the tangent space
''M'' at each point ''p''. A common convention is to take ''g'' to be smooth, which means that for any smooth coordinate chart
on ''M'', the ''n''2
are smooth function
s. In the same way, one could also consider Lipschitz
Riemannian metrics or measurable
Riemannian metrics, among many other possibilities.
The family ''g''''p''
of inner products is called a Riemannian metric (or Riemannian metric tensor)
. These terms are named after the German mathematician Bernhard Riemann
. The study of Riemannian manifolds constitutes the subject called Riemannian geometry
A Riemannian metric (tensor) makes it possible to define several geometric notions on a Riemannian manifold, such as angle
at an intersection, length of a curve
of a surface and higher-dimensional analogues (volume
, etc.), extrinsic curvature
of submanifolds, and intrinsic curvature
of the manifold itself.
In 1828, Carl Friedrich Gauss
proved his Theorema Egregium
(''remarkable theorem'' in Latin), establishing an important property of surfaces. Informally, the theorem says that the curvature of a surface
can be determined entirely by measuring distances along paths on the surface. That is, curvature does not depend on how the surface might be embedded in 3-dimensional space. See ''Differential geometry of surfaces
''. Bernhard Riemann
extended Gauss's theory to higher-dimensional spaces called manifolds in a way that also allows distances and angles to be measured and the notion of curvature to be defined, again in a way that is intrinsic to the manifold and not dependent upon its embedding in higher-dimensional spaces. Albert Einstein
used the theory of pseudo-Riemannian manifold
s (a generalization of Riemannian manifolds) to develop his general theory of relativity
. In particular, his equations for gravitation are constraint
s on the curvature of spacetime.
The tangent bundle
of a smooth manifold
assigns to each point
a vector space
called the tangent space
A Riemannian metric (by its definition) assigns to each
a positive-definite inner product
along with which comes a norm
The smooth manifold
endowed with this metric
is a Riemannian manifold, denoted
When given a system of smooth local coordinates
form a basis of the vector space
Relative to this basis, one can define metric tensor "components" at each point
One could consider these as
or as a single
matrix-valued function on
note that the "Riemannian" assumption says that it is valued in the subset consisting of symmetric positive-definite matrices.
In terms of tensor algebra
, the metric tensor
can be written in terms of the dual basis
of the cotangent bundle as
are two Riemannian manifolds, with
a diffeomorphism, then
is called an isometry if
One says that a map
not assumed to be a diffeomorphism, is a local isometry if every
has an open neighborhood
is a diffeomorphism and isometry.
Regularity of a Riemannian metric
One says that the Riemannian metric
is continuous if
are continuous when given any smooth coordinate chart
One says that
is smooth if these functions are smooth when given any smooth coordinate chart. One could also consider many other types of Riemannian metrics in this spirit.
In most expository accounts of Riemannian geometry, the metrics are always taken to be smooth. However, there can be important reasons to consider metrics which are less smooth. Riemannian metrics produced by methods of geometric analysis
, in particular, can be less than smooth. See for instance (Gromov 1999) and (Shi and Tam 2002).
Examples of Riemannian manifolds will be discussed below. A famous theorem
of John Nash
states that, given any smooth Riemannian manifold
there is a (usually large) number
and an embedding
such that the pullback
of the standard Riemannian metric on
Informally, the entire structure of a smooth Riemannian manifold can be encoded by a diffeomorphism to a certain embedded submanifold of some Euclidean space. In this sense, it is arguable that nothing can be gained from the consideration of abstract smooth manifolds and their Riemannian metrics. However, there are many natural smooth Riemannian manifolds, such as the set of rotations of three-dimensional space
and the hyperbolic space
, of which any representation as a submanifold of Euclidean space will fail to represent their remarkable symmetries and properties as clearly as their abstract presentations do.
denote the standard coordinates on
Phrased differently: relative to the standard coordinates, the local representation
is given by the constant value
This is clearly a Riemannian metric, and is called the standard Riemannian structure on
It is also referred to as Euclidean space
of dimension ''n'' and ''g''''ij''can
is also called the (canonical) Euclidean metric
be a Riemannian manifold and let
be an embedded submanifold
which is at least
Then the restriction
of ''g'' to vectors tangent along ''N'' defines a Riemannian metric over ''N''.
* For example, consider
which is a smooth embedded submanifold of the Euclidean space with its standard metric. The Riemannian metric this induces on
is called the standard metric or canonical metric on
* There are many similar examples. For example, every ellipsoid in
has a natural Riemannian metric. The graph of a smooth function
is an embedded submanifold, and so has a natural Riemannian metric as well.
be a Riemannian manifold and let
be a differentiable map. Then one may consider the pullback
, which is a symmetric 2-tensor on
is the pushforward
In this setting, generally
will not be a Riemannian metric on
since it is not positive-definite. For instance, if
is constant, then
is zero. In fact,
is a Riemannian metric if and only if
is an immersion
, meaning that the linear map
is injective for each
* An important example occurs when
is not simply-connected, so that there is a covering map
This is an immersion, and so the universal cover of any Riemannian manifold automatically inherits a Riemannian metric. More generally, but by the same principle, any covering space of a Riemannian manifold inherits a Riemannian metric.
* Also, an immersed submanifold of a Riemannian manifold inherits a Riemannian metric.
be two Riemannian manifolds, and consider the cartesian product
with the usual product smooth structure. The Riemannian metrics
naturally put a Riemannian metric
which can be described in a few ways.
* Considering the decomposition
one may define
be a smooth coordinate chart on
be a smooth coordinate chart on
is a smooth coordinate chart on
For convenience let
denote the collection of positive-definite symmetric
real matrices. Denote the coordinate representation of
and denote the coordinate representation of
Then the local coordinate representation of
A standard example is to consider the n-torus
define as the n-fold product
If one gives each copy of
its standard Riemannian metric, considering
as an embedded submanifold (as above), then one can consider the product Riemannian metric on
It is called a flat torus
Convex combinations of metrics
be two Riemannian metrics on
Then, for any number
is also a Riemannian metric on
More generally, if
are any two positive numbers, then
is another Riemannian metric.
Every smooth manifold has a Riemannian metric
This is a fundamental result. Although much of the basic theory of Riemannian metrics can be developed by only using that a smooth manifold is locally Euclidean, for this result it is necessary to include in the definition of "smooth manifold" that it is Hausdorff and paracompact. The reason is that the proof makes use of a partition of unity
The metric space structure of continuous connected Riemannian manifolds
The length of piecewise continuously-differentiable curves
is differentiable, then it assigns to each
in the vector space
the size of which can be measured by the norm
defines a nonnegative function on the interval
The length is defined as the integral of this function; however, as presented here, there is no reason to expect this function to be integrable. It is typical to suppose ''g'' to be continuous and
to be continuously differentiable, so that the function to be integrated is nonnegative and continuous, and hence the length of
is well-defined. This definition can easily be extended to define the length of any piecewise-continuously differentiable curve.
In many instances, such as in defining the Riemann curvature tensor
, it is necessary to require that ''g'' has more regularity than mere continuity; this will be discussed elsewhere. For now, continuity of ''g'' will be enough to use the length defined above in order to endow ''M'' with the structure of a metric space
, provided that it is connected.
The metric space structure
It is mostly straightforward to check the well-definedness of the function
its symmetry property
its reflexivity property
and the triangle inequality
although there are some minor technical complications (such as verifying that any two points can be connected by a piecewise-differentiable path). It is more fundamental to understand that
and hence that
satisfies all of the axioms of a metric.
The observation that underlies the above proof, about comparison between lengths measured by ''g'' and Euclidean lengths measured in a smooth coordinate chart, also verifies that the metric space topology of
coincides with the original topological space structure of
Although the length of a curve is given by an explicit formula, it is generally impossible to write out the distance function
by any explicit means. In fact, if
is compact then, even when ''g'' is smooth, there always exist points where
is non-differentiable, and it can be remarkably difficult to even determine the location or nature of these points, even in seemingly simple cases such as when
is an ellipsoid.
As in the previous section, let
be a connected and continuous Riemannian manifold; consider the associated metric space
Relative to this metric space structure, one says that a path
is a unit-speed [[geodesic if for every
there exists an interval
and such that
Informally, one may say that one is asking for
to locally 'stretch itself out' as much as it can, subject to the (informally considered) unit-speed constraint. The idea is that if
is (piecewise) continuously differentiable and
then one automatically has
by applying the triangle inequality to a Riemann sum approximation of the integral defining the length of
So the unit-speed geodesic condition as given above is requiring
to be as far from one another as possible. The fact that we are only looking for curves to ''locally'' stretch themselves out is reflected by the first two examples given below; the global shape of
may force even the most innocuous geodesics to bend back and intersect themselves.
* Consider the case that
is the circle
with its standard Riemannian metric, and
is given by
is measured by the lengths of curves along
, not by the straight-line paths in the plane. This example also exhibits the necessity of selecting out the subinterval
since the curve
repeats back on itself in a particularly natural way.
* Likewise, if
is the round sphere
with its standard Riemannian metric, then a unit-speed path along an equatorial circle will be a geodesic. A unit-speed path along the other latitudinal circles will not be geodesic.
* Consider the case that
with its standard Riemannian metric. Then a unit-speed line such as
is a geodesic but the curve
from the first example above is not.
Note that unit-speed geodesics, as defined here, are by necessity continuous, and in fact Lipschitz
, but they are not necessarily differentiable or piecewise differentiable.
The Hopf-Rinow theorem
As above, let
be a connected and continuous Riemannian manifold. The Hopf-Rinow theorem
, in this setting, says that (Gromov 1999)
* if the metric space
-Cauchy sequence converges) then
** every closed and bounded subset of
** given any
there is a unit-speed geodesic
The essence of the proof is that once the first half is established, one may directly apply the Arzelà-Ascoli theorem
, in the context of the compact metric space
to a sequence of piecewise continuously-differentiable unit-speed curves from
whose lengths approximate
The resulting subsequential limit is the desired geodesic.
The assumed completeness of
is important. For example, consider the case that
is the punctured plane
with its standard Riemannian metric, and one takes
There is no unit-speed geodesic from one to the other.
be a connected and continuous Riemannian manifold. As with any metric space, one can define the diameter of
The Hopf-Rinow theorem shows that if
is complete and has finite diameter, then it is compact. Conversely, if
is compact, then the function
has a maximum, since it is a continuous function on a compact metric space. This proves the following statement:
is complete, then it is compact if and only if it has finite diameter.
This is not the case without the completeness assumption; for counterexamples one could consider any open bounded subset of a Euclidean space with the standard Riemannian metric.
Note that, more generally, and with the same one-line proof, every compact metric space has finite diameter. However the following statement is ''false'': "If a metric space is complete and has finite diameter, then it is compact." For an example of a complete and non-compact metric space of finite diameter, consider
with the uniform metric
So, although all of the terms in the above corollary of the Hopf-Rinow theorem involve only the metric space structure of
it is important that the metric is induced from a Riemannian structure.
A Riemannian manifold ''M'' is geodesically complete if for all , the exponential map
is defined for all , i.e. if any geodesic ''γ''(''t'') starting from ''p'' is defined for all values of the parameter . The Hopf–Rinow theorem
asserts that ''M'' is geodesically complete if and only if it is complete as a metric space
If ''M'' is complete, then ''M'' is non-extendable in the sense that it is not isometric to an open proper submanifold of any other Riemannian manifold. The converse is not true, however: there exist non-extendable manifolds that are not complete.
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