Tensor analysis

A generalization of vector analysis, a part of tensor calculus studying differential (and integration) operators on the algebra $ D( M) $ of differentiable tensor fields over a differentiable manifold $ M $. Also the extension of this theory to more general geometric objects than tensor fields, such as tensor densities, vector-valued differential forms, etc. is considered as a part of tensor analysis.

References

[1]	P.K. [P.K. Rashevskii] Rashewski, "Riemannsche Geometrie und Tensoranalyse" , Deutsch. Verlag Wissenschaft. (1959) (Translated from Russian)
[2]	J.A. Schouten, "Tensor analysis for physicists" , Cambridge Univ. Press (1951)
[3]	A.J. MacConnel, "Applications of tensor analysis" , Dover, reprint (1957)
[4]	I.S. Sokolnikoff, "Tensor analysis" , Wiley (1964)

Comments

The most important operators of tensor analysis map tensor fields into tensor fields, possibly changing the type of the field.

1) The covariant derivative in the direction of a vector field $ X \in D ^ {( 0,1) } ( M) $ is a linear mapping $ \nabla _ {X} $ of the vector space $ D ^ {( 0,1) } ( M) $ into itself satisfying the following conditions:

$$ \nabla _ {fX+ gY } Z = f \nabla _ {X} Z + g \nabla _ {Y} Z,\ \ \nabla _ {X} ( fZ) = f \nabla _ {X} Z + X( f ) Z, $$

where $ X, Y, Z \in D ^ {( 0,1) } ( M) $ and $ f $ and $ g $ are smooth functions on $ M $. In this interpretation the vector field $ X $ is considered as a derivation on functions, i.e. in local coordinates $ u ^ {1} \dots u ^ {n} $ with $ X = \sum _ {i= 1 } ^ {n} \xi ^ {i} ( \partial / \partial u ^ {i} ) $, for the derivative of the function $ f $ one has $ X( f ) = \sum _ {i= 1 } ^ {n} \xi ^ {i} ( \partial f / \partial u ^ {i} ) $. The prescription of such an operator $ \nabla : D ^ {( 0,1) } ( M) \times D ^ {( 0,1) } ( M) \rightarrow D ^ {( 0,1) } ( M) $ defines a linear connection and a parallel displacement along curves on $ M $.

The extension of the covariant derivative to arbitrary tensor fields on $ M $ can be characterized by the properties that it is a derivation preserving the type of the tensor field and commuting with contraction. Regarding a tensor field of type $ ( r, s) $ as a field of multilinear mappings which are defined on $ r $ copies of the corresponding tangent space and $ s $ copies of the cotangent space, this extension is defined for $ T \in D ^ {( r,s) } ( M) $ as follows:

$$ ( \nabla _ {X} T)( X _ {1} \dots X _ {r} ,\ \omega ^ {1} \dots \omega ^ {s} ) = $$

$$ = \ \nabla _ {X} ( T( X _ {1} \dots X _ {r} , \omega ^ {1} \dots \omega ^ {s} )) - $$

$$ - \sum _ {i= 1 } ^ { r } T( X _ {1} \dots \nabla _ {X} X _ {i} \dots X _ {r} , \omega ^ {1} \dots \omega ^ {s} ) - $$

$$ - \sum _ {j= 1 } ^ { s } T( X _ {1} \dots X _ {r} , \omega ^ {1} \dots \nabla _ {X} \omega ^ {j} \dots \omega ^ {s} ), $$

where $ ( \nabla _ {X} \omega )( Y) = \nabla _ {X} ( \omega ( Y))- \omega ( \nabla _ {X} Y) $, $ X, Y, X _ {1} \dots X _ {r} \in D ^ {( 0,1) } ( M) $ and $ \omega , \omega _ {1} \dots \omega _ {s} \in D ^ {( 1,0) } ( M) $. This implies in local coordinates, using the notations introduced above and the connection coefficients $ \Gamma _ {ij } ^ {k} $ given by $ \nabla _ {\partial / \partial u ^ {i} } \partial / \partial u ^ {j} = \sum _ {k= 1 } ^ {n} \Gamma _ {ij } ^ {k} \partial / \partial u ^ {k} $, the following formula in terms of the components $ T _ {i _ {1} \dots i _ {r} } ^ {j _ {1} \dots j _ {s} } $ of $ T $:

$$ ( \nabla _ {X} T) _ {i _ {1} \dots i _ {r} } ^ {j _ {1} \dots j _ {s} } = \ \sum _ {k= 1 } ^ { n } \xi ^ {k} \left [ \frac \partial {\partial u ^ {k} } ( T _ {i _ {1} \dots i _ {r} } ^ {j _ {1} \dots j _ {s} } ) \right . - $$

$$ - \left . \sum _ {\nu = 1 } ^ { r } \sum _ {l= 1 } ^ { n } \Gamma _ {ki _ \nu } ^ {l} T _ {i _ {1} \dots i _ {\nu - 1 } l i _ {\nu + 1 } \dots i _ {r} } ^ {j _ {1} \dots j _ {s} } + \sum _ {\mu = 1 } ^ { s } \sum _ {l= 1 } ^ { n } \Gamma _ {kl } ^ {j _ \mu } T _ {i _ {1} \dots i _ {r} } ^ {j _ {1} \dots j _ {\mu - 1 } l j _ {\mu + 1 } \dots j _ {s} } \right ] . $$

From this definition one easily obtains the covariant differential $ \nabla T \in D ^ {( r+ 1,s) } ( M) $ of the tensor field $ T \in D ^ {( r,s) } ( M) $ by

$$ ( \nabla T)( X _ {0} , X _ {1} \dots X _ {r} ,\ \omega _ {1} \dots \omega _ {s} ) = $$

$$ = \ ( \nabla _ {X _ {0} } T)( X _ {1} \dots X _ {r} , \omega _ {1} \dots \omega _ {s} ), $$

where the tensorial character of the first entry can be seen from the rules for $ \nabla $ presented above. In the case of the Levi-Civita connection of a Riemannian metric on $ M $, the trace of the covariant differential of $ T $ with respect to the first and some other covariant entry leads to an extension of the divergence to tensor fields.

2) The Lie derivative along a vector field $ X $ is the mapping $ L _ {X} : D ^ {( 0,1) } ( M) \rightarrow D ^ {( 0,1) } ( M) $ defined by $ L _ {X} Y = [ X, Y] $, where $ [ X, Y]( f ) = X( Y( f )) - Y( X( f )) $ for any smooth function $ f $ on $ M $. The extension of the Lie derivative to arbitrary tensor fields $ T \in D ^ {( r,s) } ( M) $ can be defined in the same way as the covariant derivative:

$$ ( L _ {X} T)( X _ {1} \dots X _ {r} , \omega ^ {1} \dots \omega ^ {s} ) = $$

$$ = \ \nabla _ {X} ( T( X _ {1} \dots X _ {r} , \omega ^ {1} \dots \omega ^ {s} )) - $$

$$ - \sum _ {i= 1 } ^ { r } T( X _ {1} \dots L _ {X} X _ {i} \dots X _ {r} , \omega ^ {1} \dots \omega ^ {s} ) - $$

$$ - \sum _ {j= 1 } ^ { s } T( X _ {1} \dots X _ {r} , \omega ^ {1} \dots L _ {X} \omega ^ {j} \dots \omega ^ {s} ), $$

where $ ( L _ {X} \omega )( Y) = \nabla _ {X} ( \omega ( Y))- \omega ( L _ {X} Y) $, $ X, Y, X _ {1} \dots X _ {r} \in D ^ {( 0,1) } ( M) $ and $ \omega , \omega _ {1} \dots \omega _ {s} \in D ^ {( 1,0) } ( M) $. This implies in local coordinates,

$$ ( L _ {X} T) _ {i _ {1} \dots i _ {r} } ^ {j _ {1} \dots j _ {s} } = \ \sum _ {k= 1 } ^ { n } \xi ^ {k} \frac \partial {\partial u ^ {k} } ( T _ {i _ {1} \dots i _ {r} } ^ {j _ {1} \dots j _ {s} } ) + $$

$$ + \sum _ {k= 1 } ^ { n } \sum _ {\nu = 1 } ^ { r } \frac{ \partial \xi ^ {k} }{\partial u ^ {i _ \nu } } T _ {i _ {1} \dots i _ { \nu - 1 } k i _ {\nu + 1 } \dots i _ {r} } ^ {j _ {1} \dots j _ {s} } - \sum _ {k= 1 } ^ { n } \sum _ {\mu = 1 } ^ { s } \frac{ \partial \xi ^ {j _ \mu } }{\partial u ^ {k} } T _ {i _ {1} \dots i _ {r} } ^ {j _ {1} \dots j _ {\mu - 1 } k j _ {\mu + 1 } \dots j _ {s} } . $$

3) The exterior differential or exterior derivative is a linear operator $ d $ assigning to an (exterior) differential form $ \omega \in F ^ {p} ( M) $ of degree $ p $ a differential form $ d \omega $ of degree $ p+ 1 $ such that the following compatibility with the wedge product (exterior product) of differential forms is satisfied:

$$ d( \omega _ {1} \wedge \omega _ {2} ) = \ d \omega _ {1} \wedge \omega _ {2} + (- 1) ^ {p } \omega _ {1} \wedge d \omega _ {2} , $$

where $ \omega _ {i} \in F ^ {p } ( M) $. Furthermore, one assumes that for a smooth function $ f $( a differential form of degree $ 0 $), $ df $ is given by the usual differential of $ f $ and that $ ddf= 0 $. This implies $ dd \omega = 0 $ in general. For $ \omega \in F ^ {p} ( M) $, the exterior differential can be described by:

$$ ( d \omega )( X _ {1} \dots X _ {p+ 1 } ) = $$

$$ = \ \sum _ {j= 1 } ^ { p } (- 1) ^ {j+ 1 } X _ {j} \omega ( X _ {1} \dots X _ {j- 1 } , X _ {j+ 1 } \dots X _ {p+ 1 } ) + $$

$$ + \sum _ {i< j } (- 1) ^ {i+ j } \omega ([ X _ {i} , X _ {j} ], X _ {1} \dots X _ {i- 1 } , \ $$

$$ {} X _ {i+ 1 } \dots X _ {j- 1 } , X _ {j+ 1 } \dots X _ {p+ 1 } ) . $$

If in local coordinates $ \omega = \sum _ {i _ {1} \dots i _ {p} } ^ {n} \omega _ {i _ {1} \dots i _ {p} } du ^ {i _ {1} } \wedge \dots \wedge du ^ {i _ {p} } $, then

$$ d \omega = \sum _ {i _ {1} \dots i _ {p} } ^ { n } \sum _ {k= 1 } ^ { n } \frac{\partial \omega _ {i _ {1} \dots i _ {p} } }{\partial u ^ {k} } du ^ {k} \wedge du ^ {i _ {1} } \wedge \dots \wedge du ^ {i _ {p} } . $$

The fundamental operators of classical vector analysis may be described in terms of forms and exterior differentiation. For instance, the operator $ d $, acting on differential forms of degree $ 2 $ in $ \mathbf R ^ {3} $, corresponds to the operator $ \mathop{\rm rot} $( $ \mathop{\rm curl} $). Differential forms are the suitable objects for the theory of integration on manifolds. Using the exterior derivative, the general form of the Stokes theorem can be given.

4) As an example of the application of tensor analysis in differential geometry, the curvature tensor $ R $ of a linear connection should be mentioned. This tensor is of type $ ( 3, 1) $. In terms of the corresponding covariant derivative $ \nabla $ and vector fields $ X, Y, Z \in D ^ {( 0,1) } ( M) $, $ R $ is given as follows:

$$ R( X, Y) Z = \nabla _ {X} \nabla _ {Y} Z - \nabla _ {Y} \nabla _ {X} Z - \nabla _ {[ X,Y] } Z, $$

where, as usual, the contravariant entry is identified with a vector-valued multilinear mapping. In case of the Levi-Civita connection of a Riemannian metric $ g $ one obtains for the components $ R _ {ikj } ^ {l} $ of $ R $:

$$ R _ {ikj } ^ {l} = \ \frac{\partial \Gamma _ {ik } ^ {l} }{\partial u ^ {j} } - \frac{\partial \Gamma _ {ij } ^ {l} }{\partial u ^ {k} } + \sum _ {\nu = 1 } ^ { n } \Gamma _ {ik } ^ \nu \Gamma _ {\nu j } ^ {l} - \sum _ {\nu = 1 } ^ { n } \Gamma _ {ij } ^ \nu \Gamma _ {\nu k } ^ {l} , $$

where the connection coefficients are obtained from the metric as follows:

$$ \Gamma _ {jk } ^ {i} = { \frac{1}{2} } \sum _ {\nu = 1 } ^ { n } g ^ {i \nu } \left ( \frac{\partial g _ {j \nu } }{\partial u ^ {k} } - \frac{\partial g _ {jk } }{\partial u ^ \nu } + \frac{\partial g _ {\nu k } }{\partial u ^ {j} } \right ) . $$

References

[a1]	W. Klingenberg, "Riemannian geometry" , de Gruyter (1982) (Translated from German)
[a2]	N.J. Hicks, "Notes on differential geometry" , v. Nostrand (1965)
[a3]	S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 1–2 , Interscience (1963)