| Cotangent Space |
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Note that since the tangent space and the cotangent space at a point are both real vector spaces of the same dimension, they are Isomorphic to each other. However, they are ''not'' Naturally Isomorphic . That is, given a tangent covector there is no canonical tangent vector associated with it. The situation changes with the introduction of a Riemannian Metric or a Symplectic Form in which case the added structure gives rise to a natural isomorphism. For this reason it is important to maintain the distinction between the tangent space and the cotangent space. Many definitions are more natural on one space than on the other. All the cotangent spaces of a manifold can be "glued together" to form a new differentiable manifold of twice the dimension, the Cotangent Bundle of the manifold. FORMAL DEFINITIONS Definition as linear functionals Let ''M'' be a smooth manifold and let ''p'' be a point in ''M''. Let ''T''''p''''M'' be the Tangent Space at ''p''. Then the cotangent space at ''p'' is defined as the Dual Space of ''T''''p''''M'':
:φ : ''T''''p''''M''→R
Alternate definition In some cases, one might like to have a direct definition of the cotangent space without reference to the tangent space. Such a definition can be formulated in terms of Equivalence Class es of smooth functions on ''M''.
THE DIFFERENTIAL OF A FUNCTION Let ''M'' be a smooth manifold and let ''f'' ∈ C∞(''M'') be a Smooth Function . The differential of ''f'' at a point ''p'' is the map df where ''X''''p'' is a Tangent Vector at ''p'', thought of as a derivation. That is is the Lie Derivative of ''f'' in the direction ''X'', and one has . Equivalently, we can think of tangent vectors as tangents to curves, and write df In either case, ''df''''p'' is a linear map on ''T''''p''''M'' and hence it is a tangent covector at ''p''.
# ''d'' is a linear map: ''d''(''af'' + ''bg'') = ''a df'' + ''b dg'' for constants ''a'' and ''b'', # ''d''(''fg'')''p'' = ''f''(''p'')''dg'' + ''g''(''p'')''df'',
THE PULLBACK OF A SMOOTH MAP Just as every differentiable map ''f'' : ''M'' → ''N'' between manifolds induces a linear map (called the ''pushforward'' or ''derivative'') between the tangent spaces
every such map induces a linear map (called the '' Pullback '') between the cotangent spaces, only this time in the reverse direction:
The pullback is naturally defined as the dual (or transpose) of the Pushforward . Unraveling the definition, this means the following:
If we define tangent covectors in terms of equivalence classes of smooth maps vanishing at a point then the definition of the pullback is even more straightforward. Let ''g'' be a smooth function on ''N'' vanishing at ''f''(''p''). Then the pullback of the covector determined by ''g'' (denoted ''dg'') is given by
That is, it is the equivalence class of functions on ''M'' vanishing at ''p'' determined by ''g'' o ''f''. EXTERIOR POWERS
For this reason, tangent covectors are frequently called '' One-form s''. REFERENCES
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