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Riemannian manifold
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(Definition)
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A Riemannian metric tensor is a covariant, type  tensor field
 such that at each point  , the bilinear form
 is symmetric and positive definite. Here  is the cotangent bundle of  (defined as a sheaf),  is the set of
global sections of
 , and  is the value of the function  at the point  .
Let
 be a system of local coordinates on an open subset
 , let
 be the corresponding coframe of 1-forms, and let
 be the corresponding dual frame of vector fields. Using the local coordinates, the metric tensor has the unique expression
where the metric tensor components
are smooth functions on  .
Once we fix the local coordinates, the functions  completely determine the Riemannian metric. Thus, at each point  , the matrix
 is symmetric, and positive definite. Indeed, it is possible to define a Riemannian structure on a manifold  by specifying an atlas over  together with a matrix of functions  on each coordinate chart which are symmetric and positive definite,
with the proviso that the  's must be compatible with each other on overlaps.
A manifold  together with a Riemannian metric tensor  is called a Riemannian manifold.
Note: A Riemannian metric tensor on  is not a distance metric on  . However, on a connected manifold every Riemannian metric tensor on  induces a distance metric on  , given by
where the infimum is taken over all rectifiable curves
![$ c:[0,1]\to M$](http://images.planetmath.org:8080/cache/objects/3452/l2h/img33.png "$ c:[0,1]\to M$") with  and  .
Often, it is the  that are referred to as the “Riemannian metric”. This, however, is a misnomer. Properly speaking, the  should be called local coordinate components of a metric tensor, where as “Riemannian metric” should refer to the distance function defined above. However, the practice of calling the collection of  's by the misnomer “Riemannian metric” appears to have stuck.
Remark Both the Riemannian manifold and Riemannian metric tensor are fundamental concepts in Einstein's General Relativity (GR) theory where the “Riemannian metric" and curvature of the physical Riemannian space-time are changed by the presence of massive bodies and energy according to Einstein's fundamental GR equations.
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"Riemannian manifold" is owned by djao. [ full author list (3) ]
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See Also: quantum Riemannian geometry, gradient, category of Riemannian manifolds, homotopy category,  complex of spin networks, complex representation theorems
| Other names: |
Riemann space and metric |
| Also defines: |
Riemannian metric, Riemannian structure, metric tensor |
| Keywords: |
Riemannian metric tensor, Riemannian manifold, Riemann space, Quantum Riemannian Geometry |
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Cross-references: equations, bodies, curvature, theory, collection, distance, rectifiable curves, infimum, induces, connected, distance metric, compatible, coordinate chart, atlas, manifold, matrix, fix, smooth functions, components, expression, vector fields, frame, 1-forms, coframe, open subset, local coordinates, function, global sections, sheaf, cotangent bundle, positive definite, symmetric, bilinear form, point, field, type, tensor
There are 54 references to this entry.
This is version 21 of Riemannian manifold, born on 2002-09-12, modified 2008-10-11.
Object id is 3452, canonical name is RiemannianMetric.
Accessed 16057 times total.
Classification:
| AMS MSC: | 53B20 (Differential geometry :: Local differential geometry :: Local Riemannian geometry) | | | 53B21 (Differential geometry :: Local differential geometry :: Methods of Riemannian geometry) |
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Pending Errata and Addenda
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![$\displaystyle d(x,y) := \inf \left\{ \int_0^1 \left[g\!\!\left( \frac{dc}{dt}, \frac{dc}{dt}\right)_{\!c(t)}\right]^{1/2}dt \right\} ,\quad x,y\in M,$](http://images.planetmath.org:8080/cache/objects/3452/l2h/img32.png "$\displaystyle d(x,y) := \inf \left\{ \int_0^1 \left[g\!\!\left( \frac{dc}{dt}, \frac{dc}{dt}\right)_{\!c(t)}\right]^{1/2}dt \right\} ,\quad x,y\in M,$")