Εφαπτόμενη Ινοδέσμη
- Ένα είδος Ινοδέσμης
Ετυμολογία[]
Η ονομασία "Ινοδέσμη" σχετίζεται ετυμολογικά με την λέξη "δέσμη".
Εισαγωγή[]
In differential geometry, the tangent bundle of a differentiable manifold is a manifold which assembles all the tangent vectors in . As a set, it is given by the disjoint union
The disjoint union ensures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector.
This is graphically illustrated in the accompanying picture for tangent bundle of circle S1,
see Examples section: all tangents to a circle lie in the plane of the circle.
In order to make them disjoint it is necessary to align them in a plane perpendicular to the plane of the circle. of the tangent spaces of . That is,
where denotes the tangent space to at the point . So, an element of can be thought of as a pair , where is a point in and is a tangent vector to at .
So, an element of can be thought of\as a pair , where is a point in and is a tangent vector to at . There is a natural projection
defined by . This projection maps each tangent space to the single point .
The tangent bundle comes equipped with a natural topology (described in a section below). With this topology, the tangent bundle to a manifold is the prototypical example of a vector bundle (a fiber bundle whose fibers are vector spaces).
A section of is a vector field on , and the dual bundle to is the cotangent bundle, which is the disjoint union of the cotangent spaces of .
By definition, a manifold is parallelizable if and only if the tangent bundle is trivial.
Υποσημειώσεις[]
Εσωτερική Αρθρογραφία[]
- Hopf bundle
- I-bundle
- Principal bundle
- Trivial bundle
- Pullback bundle
- Universal bundle
- Vector bundle
- Affine bundle
- Equivariant bundle
- Fibred manifold
- Trivialization
- Quasifibration
- Covering map
- Fibration
- Gauge theory
Βιβλιογραφία[]
Ιστογραφία[]
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