Tensor of Killing-Yano

In Geometry riemannienne, a tensor of Killing-Yano is a generalization of the concept of Vecteur of Killing to a Tenseur of higher size. They were introduced in 1952 by Kentarô Yano. An antisymmetric tensor of order p $f_ {a_1 a_2… a_p}$ is known as of Killing-Yano when it checks the equation

D_b f_ {turnover. _2… a_p} + D_c f_ {B a_2… a_p} = 0 \,

This equation differs from the usual generalization of the concept of vector of Killing to tensors of a nature more raised, called tensor of Killing by what the Dérivée covariante D is symmetrized with only one index of the tensor and not the totality of those, as it is the case for the tensors of Killing.

Commonplace tensors of Killing-Yano

All Vecteur of Killing is a tensor of Killing of order 1 and one tensor of Killing-Yano.

The tensor antisymmetric completion (known as of Levi-Civita) $\ epsilon_ {a_1 a_2… a_n}$ , where N is the dimension of the variety is a tensor of Killing-Yano, its derivative covariante being always null (see Nullité of derived the covariante tensor dualisor).

Construction of tensors of Killing starting from tensors of Killing-Yano

There exist several ways of building tensors of Killing (symmetrical) starting from tensors of Killing-Yano.

First of all, two commonplace tensors of Killing can be obtained starting from tensors of Killing-Yano:

Starting from a tensor of Killing-Yano of order 1 $\ xi_a$ , one can build a tensor of Killing $K_ {ab}$ of order of 2 according to

K_ {ab} = \ xi_a \ xi_b

Starting from the completely antisymmetric tensor $\ epsilon_ {a_1 a_2… a_n}$ , one can build the tensor of commonplace Killing

K_ {ab} = \ epsilon_ {B a_2… a_n} \ epsilon^ {a_2… a_n C} g_ {Ca} = - 6 g_ {ab}

In a more interesting way, starting from two tensors of Killing-Yano of order 2 $A_ {ab}$ and $B_ {ab}$ , one can build the tensor of Killing of a nature 2 $K_ {ab}$ according to

K_ {ab} = g^ {Cd} \ left (A_ {ac} B_ {dB} + B_ {ac} A_ {dB} \ right)

Starting from a tensor of Killing-Yano of order N -1, $A_ {a_2… a_n}$ , one can build the associated vector within the meaning of Hodge (see Dualité of Hodge),

A^a = \ epsilon^ {has a_2… a_n} A_ {a_2… a_n}

Owing to the fact that the tensor

A_ {a_2… a_n}

is of Killing-Yano, the vector has is not Killing-Yano, but obeys the equation

D_a A_b = \ frac {1} {N} g_ {ab} D_c. A. ^c

This property allows of built a tensor of Killing

K_ {ab}

starting from two such vectors, defined by:

K_ {ab} = A_a B_b + A_b B_a - 2 A^c B_c g_ {ab}

Any combination linéraire of tensors of Killing-Yano is also a tensor of Killing-Yano.

Properties

A certain number of properties of the four-dimensional space times implying the tensors of Killing-Yano were exhibées by C.D. Collinson and H. Stephani in the current of the Années 1970 , , .

If a space time admits a tensor of Killing-Yano not degenerated, then this one can be written in the form

A_ {ab} = X (l_a k_b - k_a l_b) + I Y (m_a \ bar m_b - \ bar m_a m_b)

where K , L , m and

\ bar m

forms a tetrad and the functions X and Y obey a certain number of differential equations. Moreover, the tensor of Killing-Yano obeys the following relation with the Tenseur of Ricci

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