In
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 mult ...
, a Riemannian manifold or Riemannian space , so called after the German mathematician
Bernhard Riemann
Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first ...
, is a
real,
smooth manifold ''M'' equipped with a positive-definite
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 ...
''g''
''p'' on the
tangent space ''T''
''p''''M'' at each point ''p''.
The family ''g''
''p'' of inner products is called a
Riemannian metric (or Riemannian metric tensor).
Riemannian geometry is the study of Riemannian manifolds.
A common convention is to take ''g'' to be
smooth, which means that for any smooth
coordinate chart on ''M'', the ''n''
2 functions
:
are
smooth function
In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if ...
s. These functions are commonly designated as
.
With further restrictions on the
, one could also consider
Lipschitz Lipschitz, Lipshitz, or Lipchitz, is an Ashkenazi Jewish (Yiddish/German-Jewish) surname. The surname has many variants, including: Lifshitz ( Lifschitz), Lifshits, Lifshuts, Lefschetz; Lipschitz, Lipshitz, Lipshits, Lopshits, Lipschutz (Lip ...
Riemannian metrics or
measurable Riemannian metrics, among many other possibilities.
A Riemannian metric (tensor) makes it possible to define several geometric notions on a Riemannian manifold, such as
angle
In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle.
Angles formed by two rays lie in the plane that contains the rays. Angles ...
at an intersection, length of a
curve,
area
Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an op ...
of a surface and higher-dimensional analogues (
volume
Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). ...
, etc.),
extrinsic curvature of submanifolds, and
intrinsic curvature of the manifold itself.
Introduction
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
Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first ...
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
Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theor ...
used the theory of
pseudo-Riemannian manifolds (a generalization of Riemannian manifolds) to develop his
general theory of relativity. In particular, his equations for gravitation are
constraints on the curvature of spacetime.
Definition
The
tangent bundle of a
smooth manifold assigns to each point
of
a vector space
called the
tangent space of
at
A Riemannian metric (by its definition) assigns to each
a positive-definite inner product
along with which comes a norm
defined by
The
smooth manifold endowed with this metric
is a Riemannian manifold, denoted
.
When given a system of smooth
local coordinates on
given by
real-valued functions
the vectors
:
form a basis of the vector space
for any
Relative to this basis, one can define metric tensor "components" at each point
by
:
One could consider these as
individual functions
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
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 allow ...
can be written in terms of the
dual basis of the cotangent bundle as
:
Isometries
If
and
are two Riemannian manifolds, with
a
diffeomorphism, then
is called an isometry if
i.e. if
:
for all
and
One says that a map
not assumed to be a diffeomorphism, is a local isometry if every
has an open neighborhood
such that
is an isometry (and thus a diffeomorphism).
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).
Overview
Examples of Riemannian manifolds will be discussed below. A famous
theorem
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of t ...
of
John Nash states that, given any smooth Riemannian manifold
there is a (usually large) number
and an embedding
such that 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: ...
by
of the standard Riemannian metric on
is
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.
Examples
Euclidean space
Let
denote the standard coordinates on
Then define
by
:
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
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
of dimension ''n'' and ''g''
''ij''can is also called the (canonical)
Euclidean metric.
Embedded submanifolds
Let
be a Riemannian manifold and let
be an
embedded submanifold of
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.
Immersions
Let
be a Riemannian manifold and let
be a differentiable map. Then one may consider 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: ...
of
via
, which is a symmetric 2-tensor on
defined by
:
where
is the
pushforward of
by
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.
Product metrics
Let
and
be two Riemannian manifolds, and consider the cartesian product
with the usual product smooth structure. The Riemannian metrics
and
naturally put a Riemannian metric
on
which can be described in a few ways.
* Considering the decomposition
one may define
:
* Let
be a smooth coordinate chart on
and let
be a smooth coordinate chart on
Then
is a smooth coordinate chart on
For convenience let
denote the collection of positive-definite symmetric
real matrices. Denote the coordinate representation of
relative to
by
and denote the coordinate representation of
relative to
by
Then the local coordinate representation of
relative to
is
given by
::
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
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does not tou ...
.
Convex combinations of metrics
Let
and
be two Riemannian metrics on
Then, for any number
:
is also a Riemannian metric on
More generally, if
and
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
If
is differentiable, then it assigns to each
a vector
in the vector space
the size of which can be measured by the norm
So
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
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 setti ...
, provided that it is connected.
The metric space structure
Precisely, define
by
:
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
ensures
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.
Geodesics
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
which contains
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
for all
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
and
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
Recall that
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
is
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 Lipschitz, Lipshitz, or Lipchitz, is an Ashkenazi Jewish (Yiddish/German-Jewish) surname. The surname has many variants, including: Lifshitz ( Lifschitz), Lifshits, Lifshuts, Lefschetz; Lipschitz, Lipshitz, Lipshits, Lopshits, Lipschutz (Lip ...
, 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
is
complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
(i.e. every
-Cauchy sequence converges) then
** every closed and bounded subset of
is compact.
** given any
there is a unit-speed geodesic
from
to
such that
for all
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
to
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
and
There is no unit-speed geodesic from one to the other.
The diameter
Let
be a connected and continuous Riemannian manifold. As with any metric space, one can define the diameter of
to be
:
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:
* If
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.
Riemannian metrics
Geodesic completeness
A Riemannian manifold ''M'' is geodesically complete if for all , the
exponential map exp
''p'' 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.
Infinite-dimensional manifolds
The statements and theorems above are for finite-dimensional manifolds—manifolds whose charts map to open subsets of
These can be extended, to a certain degree, to infinite-dimensional manifolds; that is, manifolds that are modeled after a
topological vector space; for example,
Fréchet,
Banach and
Hilbert manifolds.
Definitions
Riemannian metrics are defined in a way similar to the finite-dimensional case. However there is a distinction between two types of Riemannian metrics:
* A weak Riemannian metric on
is a smooth function
such that for any
the restriction
is an inner product on
* A strong Riemannian metric on
is a weak Riemannian metric, such that
induces the topology on
Note that if
is not a Hilbert manifold then
cannot be a strong metric.
Examples
* If
is a
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
, then for any
one can identify
with
By setting for all
one obtains a strong Riemannian metric.
* Let
be a compact Riemannian manifold and denote by
its diffeomorphism group. It is a smooth manifold (
see here) and in fact, a
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addi ...
. Its tangent bundle at the identity is the set of smooth vector fields on
Let
be a
volume form In mathematics, a volume form or top-dimensional form is a differential form of degree equal to the differentiable manifold dimension. Thus on a manifold M of dimension n, a volume form is an n-form. It is an element of the space of sections of th ...
on
Then one can define
the
weak Riemannian metric, on
Let
Then for
and define
The
weak Riemannian metric on
induces vanishing geodesic distance, see Michor and Mumford (2005).
The metric space structure
Length of curves is defined in a way similar to the finite-dimensional case. The function