Covariant

of a tensor $ t $ on a finite-dimensional vector space $ V $

A mapping $ \phi $ of the space $ T $ of tensors of a fixed type over $ V $ into a space $ S $ of covariant tensors over $ V $ such that $ \phi ( g ( t) ) = g ( \phi ( t) ) $ for any non-singular linear transformation $ g $ of $ V $ and any $ t \in T $. This is the definition of the covariant of a tensor with respect to the general linear group $ \mathop{\rm GL} ( V) $. If $ g $ is not arbitrary but belongs to a fixed subgroup $ G \subset \mathop{\rm GL} ( V ) $, then one obtains the definition of a covariant of a tensor relative to $ G $, or simply a covariant of $ G $.

In coordinate language, a covariant of a tensor on a finite-dimensional vector space is a set of functions

$$ s _ {i} = \phi _ {i} ( t _ {1} \dots t _ {n} ) ,\ \ i = 1 \dots m , $$

of the components of the tensor $ t $ with the following properties: Under a change of the set of numbers $ t _ {1} \dots t _ {n} $ defined by a non-singular linear transformation $ g \in G $, the set of numbers $ s _ {1} \dots s _ {m} $ changes according to that of a covariant tensor $ s $ over $ V $ under the transformation $ g $. In similar fashion one defines (by considering instead of one tensor $ t $ a finite collection of tensors) a joint covariant of the system of tensors. If instead, one replaces the covariance condition for the tensor $ s $ by contravariance, one obtains the notion of a contravariant.

The notion of a covariant arose in the classical theory of invariants and is a special case of the notion of a comitant. The components of any tensor can be regarded as the coefficients of an appropriate form in several contravariant and covariant vectors (that is, vectors of $ V $ and its dual $ V ^ {*} $, cf. "form associated to a tensor" in the article Tensor on a vector space). Suppose that the form $ f $ corresponds in this manner to the tensor $ t $ and that the form $ h $ corresponds to its covariant $ s $. Then $ h $ if a form of contravariant vectors only. In the classical theory of invariants $ h $ was called the covariant of $ f $. A case that was particularly often considered is when $ h $ is a form in one single contravariant vector. The degree of this form is called the order of the covariant. If the coefficients of $ h $ are polynomials in the coefficients of $ f $, then the highest of the degrees of these polynomials is called the degree of the covariant.

Example. Let $ f = \sum a _ {i _ {1} \dots i _ {r} } x ^ {i _ {1} } \dots x ^ {i _ {r} } $ be a form of a degree $ r $, where $ x ^ {1} \dots x ^ {n} $ are the components of a contravariant vector. The form $ f $ corresponds to a symmetric covariant tensor $ t $ of valency $ r $ with components $ a _ {i _ {1} \dots i _ {r} } $. Let

$$ h = \frac{1}{r ^ {n} ( r - 1 ) ^ {n} } \left | \begin{array}{ccc} \frac{\partial ^ {2} f }{( \partial x ^ {1} ) ^ {2} } &\dots & \frac{\partial ^ {2} f }{\partial x ^ {1} \partial x ^ {n} } \\ \dots &\dots &\dots \\ \frac{\partial ^ {2} f }{\partial x ^ {n} \partial x ^ {1} } &\dots & \frac{\partial ^ {2} f }{( \partial x ^ {n} ) ^ {2} } \\ \end{array} \right | . $$

Then the coefficients of $ h $ are the components of some covariant tensor $ s $. The tensor $ s $( or the form $ h $) is a covariant of the tensor $ t $( or form $ f $). The form $ h $ is called the Hessian of $ f $.

References

[1]	G.B. Gurevich, "Foundations of the theory of algebraic invariants" , Noordhoff (1964) (Translated from Russian)