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Differential geometry Definition

Affine connection

Builds on Smooth manifoldTangent bundleConnection on a vector bundleLie bracket of vector fields

Definition

Let MM be a smooth manifold with tangent bundle TMTM, and let 𝔛(M)\mathfrak{X}(M) be the space of smooth vector fields on MM. An affine connection on MM is a connection on the vector bundle TMTM. That is, it is a map βˆ‡:𝔛(M)×𝔛(M)→𝔛(M)\nabla : \mathfrak{X}(M) \times \mathfrak{X}(M) \to \mathfrak{X}(M), (X,Y)β†¦βˆ‡XY(X, Y) \mapsto \nabla_X Y, that is C∞(M)C^\infty(M)-linear in XX, ℝ\R-linear in YY, and satisfies βˆ‡X(fY)=(Xf)Y+fβˆ‡XY\nabla_X (fY) = (Xf) Y + f \nabla_X Y for every smooth function ff.

Remarks

Example: on ℝn\R^n with Y=βˆ‘jYjβˆ‚jY = \sum_j Y^j \partial_j, the formula βˆ‡XY=βˆ‘j(XYj)β€‰βˆ‚j\nabla_X Y = \sum_j (X Y^j) \, \partial_j defines an affine connection, the flat connection; the three conditions follow from the linearity and the product rule of directional derivatives. Non-example: (X,Y)↦[X,Y](X, Y) \mapsto [X, Y] is not an affine connection, since [fX,Y]=f[X,Y]βˆ’(Yf)X[fX, Y] = f [X, Y] - (Yf) X is not C∞(M)C^\infty(M)-linear in XX. In a chart, βˆ‡\nabla is given by its Christoffel symbols Ξ“ijk\Gamma^k_{ij}, defined by βˆ‡βˆ‚iβˆ‚j=Ξ“ijkβˆ‚k\nabla_{\partial_i} \partial_j = \Gamma^k_{ij} \partial_k. Every smooth manifold has affine connections: glue the flat connections of charts with a partition of unity. Two important tensors measure an affine connection: its torsion βˆ‡XYβˆ’βˆ‡YXβˆ’[X,Y]\nabla_X Y - \nabla_Y X - [X, Y], built with the Lie bracket, and its curvature.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Connection on a vector bundle, concept, no evidence levelConnection on a vecto…conceptLie bracket of vector fields, concept, no evidence levelLie bracket of vector…conceptSmooth manifold, concept, no evidence levelSmooth manifoldconceptTangent bundle, concept, no evidence levelTangent bundleconceptAffine connection, definition, no evidence levelAffine connectiondefinition
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  1. Connection on a vector bundle concept
  2. Lie bracket of vector fields concept
  3. Smooth manifold concept
  4. Tangent bundle concept
  5. Affine connection definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Affine_connection
About this page
TagDWYPS58A7Q
URIm2://DWYPS58A7Q
Kinddefinition
WritingW0 stub
MSC53B05
DefinesAffine connection
UsesSmooth manifold, Tangent bundle, Connection on a vector bundle, Lie bracket of vector fields
Changed2 November 2026

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