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| In [[mathematics]], the '''rank''' of a [[differentiable map]] ''f'' : ''M'' → ''N'' between [[differentiable manifold]]s at a point ''p'' ∈ ''M'' is the [[rank (linear algebra)|rank]] of the [[pushforward (differential)|derivative]] of ''f'' at ''p''. Recall that the derivative of ''f'' at ''p'' is a [[linear map]]
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| :<math>d_p f : T_p M \to T_{f(p)}N\,</math> | |
| from the [[tangent space]] at ''p'' to the tangent space at ''f''(''p''). As a linear map between [[vector space]]s it has a well-defined rank, which is just the [[dimension]] of the [[image (mathematics)|image]] in ''T''<sub>''f''(''p'')</sub>''N'':
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| :<math>\operatorname{rank}(f)_p = \dim(\operatorname{im}(T_p f)).</math>
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| ==Constant rank maps==
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| A differentiable map ''f'' : ''M'' → ''N'' is said to have '''constant rank''' if the rank of ''f'' is the same for all ''p'' in ''M''. Constant rank maps have a number of nice properties and are an important concept in [[differential topology]].
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| Three special cases of constant rank maps occur. A constant rank map ''f'' : ''M'' → ''N'' is
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| *an [[immersion (mathematics)|immersion]] if rank ''f'' = dim ''M'' (i.e. the derivative is everywhere [[injective]]),
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| *a [[submersion (mathematics)|submersion]] if rank ''f'' = dim ''N'' (i.e. the derivative is everywhere [[surjective]]),
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| *a [[local diffeomorphism]] if rank ''f'' = dim ''M'' = dim ''N'' (i.e. the derivative is everywhere [[bijective]]).
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| The map ''f'' itself need not be injective, surjective, or bijective for these conditions to hold, only the behavior of the derivative is important. For example, there are injective maps which are not immersions and immersions which are not injections. However, if ''f'' : ''M'' → ''N'' is a smooth map of constant rank then
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| * if ''f'' is injective it is an immersion,
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| * if ''f'' is surjective it is a submersion,
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| * if ''f'' is bijective it is a [[diffeomorphism]].
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| Constant rank maps have a nice description in terms of [[local coordinates]]. Suppose ''M'' and ''N'' are smooth manifolds of dimensions ''m'' and ''n'' respectively, and ''f'' : ''M'' → ''N'' is a smooth map with constant rank ''k''. Then for all ''p'' in ''M'' there exist coordinates (''x''<sup>1</sup>, ..., ''x''<sup>''m''</sup>) centered at ''p'' and coordinates (''y''<sup>1</sup>, ..., ''y''<sup>''n''</sup>) centered at ''f''(''p'') such that ''f'' is given by
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| :<math>f(x^1,\ldots,x^m) = (x^1,\ldots, x^k,0,\ldots,0)\,</math> | |
| in these coordinates.
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| ==Examples==
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| [[Image:Rotating gimbal-xyz.gif|thumb|300px|[[Gimbal lock]] occurs because the map ''T''<sup>3</sup> → '''RP'''<sup>3</sup> does not have rank 3 at all points. This animation shows a set of three gimbals mounted together to allow ''three'' degrees of freedom generically (rank 3 at regular points). When all three gimbals are lined up (in the same plane), the system can only move in two dimensions from this configuration, not three – it has rank 2 at such a singular point – and is in ''gimbal lock''. In this case it can pitch or yaw, but not roll (rotate in the plane that the axes all lie in).]]
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| Maps whose rank is generically maximal, but drops at certain singular points, occur frequently in [[coordinate system]]s. For example, in [[spherical coordinates]], the rank of the map from the two angles to a point on the sphere (formally, a map ''T''<sup>2</sup> → ''S''<sup>2</sup> from the [[torus]] to the sphere) is 2 at regular points, but is only 1 at the north and south poles ([[zenith]] and [[nadir]]).
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| A subtler example occurs in [[charts on SO(3)]], the [[Rotation group SO(3)|rotation group]]. This group occurs widely in engineering, due to 3-dimensional rotations being heavily used in [[navigation]], [[nautical engineering]], and [[aerospace engineering]], among many other uses. Topologically, SO(3) is the [[real projective space]] '''RP'''<sup>3</sup>, and it is often desirable to represent rotations by a set of three numbers, known as [[Euler angles]] (in numerous variants), both because this is conceptually simple, and because one can build a combination of three [[gimbal]]s to produce rotations in three dimensions. Topologically this corresponds to a map from the 3-torus ''T''<sup>3</sup> of three angles to the real projective space '''RP'''<sup>3</sup> of rotations, but this map does not have rank 3 at all points (formally because it cannot be a [[covering map]], as the only (non-trivial) covering space is the hypersphere ''S''<sup>3</sup>), and the phenomenon of the rank dropping to 2 at certain points is referred to in engineering as ''[[gimbal lock]].''
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| ==References== | |
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| *{{cite book | first = John | last = Lee | year = 2003 | title = Introduction to Smooth Manifolds | series = Graduate Texts in Mathematics '''218''' | location = New York | publisher = Springer | isbn = 978-0-387-95495-0}}
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| [[Category:Smooth functions]]
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