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Mathematics for Physics

150 Smooth Manifold

A set of points that can be covered by mappings to open sets of real numbers of fixed dimension that is used to do calculus on curved domains.

definition [d] (Smooth Manifold = Differentiable Manifold = \(C^{\infty}\) Manifold) An \(n\)-dimensional Hausdorff, second-countable topological space covered by charts \(\{U, \phi_{U}\}\) whose overlap transition maps are \(C^{\infty}\):

  • \(M^{n}\) with atlas whose \(f_{VU}\) are infinitely differentiable .

where

  • \(M^{n}\) is the manifold of dimension \(n\).
  • \(\{U, \phi_{U}\}\) is a collection of coordinate charts.
  • \(U\) is an open set in \(M\).
  • \(\phi_{U}\) is the coordinate map on \(U\).
  • \(f_{VU}\) is the transition map between overlapping charts.
  • \(C^{\infty}\) means infinitely differentiable.

Note:

  • locally each point has a neighborhood looking like an open set in \(\mathbb{R}^{n}\).
  • equivalently, points are connected smoothly so each neighborhood looks like \(m\)-dimensional Cartesian space.

definition [d] (Smooth Manifold = Differentiable Manifold = \(C^{\infty}\) Manifold) A topological manifold \(M\) with an atlas \(\{(U_{\alpha}, \phi_{\alpha})\}\) such that every transition map

  • \(\phi_{\beta} \circ \phi_{\alpha}^{-1} : \phi_{\alpha}(U_{\alpha} \cap U_{\beta}) \rightarrow \phi_{\beta}(U_{\alpha} \cap U_{\beta})\)

is of class \(C^{\infty}\).

where

  • \(M\) is the manifold.
  • \((U_{\alpha}, \phi_{\alpha})\) is a coordinate chart.
  • \(\phi_{\alpha}: U_{\alpha} \rightarrow \mathbb{R}^{n}\) maps an open set of \(M\) into Euclidean space.
  • \(\phi_{\beta} \circ \phi_{\alpha}^{-1}\) is the transition map on the overlap.
  • \(C^{\infty}\) means infinitely differentiable.
  • \(\mathbb{R}^{n}\) is \(n\)-dimensional Euclidean space.

Note:

  • \(C^{\infty}\) transitions make derivatives of functions and tensors chart-independent.

150.1 Elementary Example

150.1.1 Simple

A smooth manifold has charts with \(C^{\infty}\) transition maps. The line \(\mathbb{R}\) with the identity chart is smooth.

\[ M = \mathbb{R},\quad n = 1 \]

\[ \phi : M \rightarrow \mathbb{R},\quad \phi(x) = x \]

where

  • \(\phi\) is a coordinate chart.
  • the identity transition is \(C^{\infty}\).

150.1.2 General

On \(\mathbb{R}^{2}\), two overlapping charts with smooth overlap give a smooth atlas.

\[ M = \mathbb{R}^{2} \]

\[ \phi(x,y) = (x,y),\quad \psi(x,y) = (x+1,y) \]

\[ \psi \circ \phi^{-1}(u,v) = (u+1,v) \]

where

  • \(\phi, \psi\) are charts.
  • \(\psi \circ \phi^{-1}\) is a \(C^{\infty}\) transition map.

150.2 References

  1. Frankel, T. The Geometry of Physics, 3rd ed. Cambridge University Press. — \(C^{\infty}\) manifold; Hausdorff, second-countable; smooth transition maps \(f_{VU}\).
  2. Hassani, S. Mathematical Physics, 2nd ed. Springer. — differentiable manifold; local Euclidean neighborhoods.
  3. Arfken, G. B., Weber, H. J., & Harris, F. E. Mathematical Methods for Physicists, 7th ed. Elsevier / Academic Press, 2013. — manifolds and curvilinear coordinate patches in applications.