In differential geometry , a branch of mathematics , the cotangential space is a vector space that is assigned to a point of a differentiable manifold . It is the dual space of the corresponding tangent space .
definition
Let be a differentiable manifold and its tangent space at the point . Then the cotangent space is defined as the dual space of . That means, the cotangent space consists of all linear forms on the tangent space .
Alternative definition
In the following another approach is shown in which the dual space is defined directly, without reference to the tangent space.
Let it be a -dimensional differentiable manifold. Further
the set of all smooth curves is through
and the set of all smooth functions that are defined in a neighborhood of :
-
.
One denotes the following equivalence relation
-
Environment of with ,
then the factor space is the vector space of the seeds over . over
a formal pairing is then defined which is linear in the first component. Now is
a linear subspace of , more precisely the null space with respect to and
is the -dimensional cotangent space in the point . One also writes for the cotangent vector .
Relation to the tangential space
With the above definition one can define an equivalence relation as follows:
The factor space
just describes the -dimensional tangent space .
If now form a basis of , one can choose a representative for each basis vector . is a differentiable map and for each one can have a curve
define, where the -th unit vector is im . Because of
are and dual to each other and you write for too .
Justification of the spellings
Be , , any function and for the curves , where the canonical basis vectors. Then in the above notations:
Thus the spelling is justified.
The linear mapping is also the total differential . So the spelling is also justified.
literature
- John M. Lee: Introduction to Smooth Manifolds (= Graduate Texts in Mathematics 218). Springer-Verlag, New York NY et al. 2003, ISBN 0-387-95448-1 .
- R. Abraham, Jerrold E. Marsden , T. Ratiu: Manifolds, tensor analysis, and applications (= Applied mathematical sciences 75). 2nd Edition. Springer, New York NY et al. 1988, ISBN 0-387-96790-7 .