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Mathematics for Physics

158 Bundle

A set that locally looks like a Cartesian product of a base set and a fiber set that is used to represent fields of vectors attached to each point.

definition [d] (Bundle = Fibre Bundle = Fiber Bundle) A fiber manifold \(F^{k}\), total space \(E\), and base \(M^{n}\), together with a projection \(\pi: E \rightarrow M^{n}\) that is locally trivial: every point of \(M\) has a neighborhood \(U\) with

  • \(\pi^{-1}(U) \cong U \times F\) .

where

  • \(F^{k}\) is the typical fiber, of dimension \(k\).
  • \(E\) is the total space.
  • \(M^{n}\) is the base manifold, of dimension \(n\).
  • \(\pi: E \rightarrow M^{n}\) is the projection map.
  • \(U\) is an open neighborhood in \(M\).
  • \(\pi^{-1}(U)\) is the preimage of \(U\) under \(\pi\).

Note:

  • \(E_{p} = \pi^{-1}(p)\) is the fiber over \(p\).
  • globally \(E\) need not equal the product \(M \times F\).

definition [d] (Bundle = Tensor Bundle = Bundle of Tensors) The bundle of type-\((r,s)\) tensors over \(M\),

  • \(T^{r}_{\ s}(M) = \bigcup_{P \in M} T^{r}_{\ s,P}(M)\) ,

with projection mapping each tensor to its base point \(P\).

where

  • \(M\) is a smooth manifold.
  • \(P\) is a point of \(M\).
  • \(T^{r}_{\ s,P}(M)\) is the space of type-\((r,s)\) tensors at \(P\).
  • \(T^{r}_{\ s}(M)\) is the tensor bundle of type \((r,s)\).
  • \(r\) is the contravariant degree.
  • \(s\) is the covariant degree.

Note:

  • a smooth section of \(T^{r}_{\ s}(M)\) is a smooth type-\((r,s)\) tensor field.
  • special cases: \(T(M) = T^{1}_{\ 0}(M)\) is the tangent bundle; \(T^{*}(M) = T^{0}_{\ 1}(M)\) is the cotangent bundle.

158.1 Elementary Example

158.1.1 Simple

A trivial bundle is the product of a base and a fiber. The projection \(\pi\) maps each pair to its base point.

\[ \pi : E \rightarrow M \]

\[ M = \{ 1,\ 2,\ 3 \} \]

\[ F = \{ a,\ b \} \]

\[ E = M \times F = \{ (1,a),\ (1,b),\ (2,a),\ (2,b),\ (3,a),\ (3,b) \} \]

\[ \pi(p,v) = p,\quad \pi^{-1}(2) = \{ (2,a),\ (2,b) \} \]

where

  • \(M\) is the base set.
  • \(F\) is the fiber set.
  • \(E\) is the total space.
  • \(\pi\) is the projection map.
  • \(\pi^{-1}(2)\) is the fiber over the point \(2\).

158.1.2 General

Local triviality means every base point has a neighborhood \(U\) with \(\pi^{-1}(U) \cong U \times F\). Here two charts cover a four-point base.

\[ M = \{ 1,\ 2,\ 3,\ 4 \} \]

\[ U = \{ 1,\ 2 \},\quad W = \{ 3,\ 4 \} \]

\[ F = \{ a,\ b,\ c \} \]

\[ \pi^{-1}(U) = U \times F,\quad \pi^{-1}(W) = W \times F \]

where

  • \(U\) and \(W\) are neighborhoods covering \(M\).
  • \(\pi^{-1}(U) \cong U \times F\) is Frankel’s local product condition.

158.2 References

  1. Frankel, T. The Geometry of Physics, 3rd ed. Cambridge University Press. — fiber bundles; local triviality; tensor bundles.
  2. Hassani, S. Mathematical Physics, 2nd ed. Springer. — \(T^{r}_{s}(M)=\bigcup_{P} T^{r}_{s,P}(M)\).
  3. Arfken, G. B., Weber, H. J., & Harris, F. E. Mathematical Methods for Physicists, 7th ed. Elsevier / Academic Press, 2013. — tensor fields on coordinate patches in the component view.