Affinor

Affinor - rank tensor $(1,1)$ ${\ displaystyle (1,1)}$ ${\ displaystyle (1,1)}$ . Designated as $\Phi _{k}^{i}$ ${\ displaystyle \ Phi _ {k} ^ {i}}$ ${\ displaystyle \ Phi _ {k} ^ {i}}$ or how $\Phi$ ${\ displaystyle \ Phi}$ $\ Phi$ . The affinor invariant is the quantity $I(\Phi )=\det \lVert \Phi _{k}^{i}\rVert$ ${\ displaystyle I (\ Phi) = \ det \ lVert \ Phi _ {k} ^ {i} \ rVert}$ ${\ displaystyle I (\ Phi) = \ det \ lVert \ Phi _ {k} ^ {i} \ rVert}$ . Affinor trace on vector $x$ ${\ displaystyle x}$ $x$ there is a vector $y^{i}=\Phi _{k}^{i}x^{k}$ ${\ displaystyle y ^ {i} = \ Phi _ {k} ^ {i} x ^ {k}}$ ${\ displaystyle y ^ {i} = \ Phi _ {k} ^ {i} x ^ {k}}$ . An affinor is called nonsingular if the equation $\Phi x=0$ ${\ displaystyle \ Phi x = 0}$ ${\ displaystyle \ Phi x = 0}$ only has zero solutions. The affinor product is the operation of incomplete coagulation of two affinors $\mathrm {X} _{j}^{i}=\Phi _{k}^{i}\Psi _{j}^{k}$ ${\ displaystyle \ mathrm {X} _ {j} ^ {i} = \ Phi _ {k} ^ {i} \ Psi _ {j} ^ {k}}$ ${\ displaystyle \ mathrm {X} _ {j} ^ {i} = \ Phi _ {k} ^ {i} \ Psi _ {j} ^ {k}}$ .

Properties

In order for the vectors to be linearly dependent, it is necessary and sufficient that the traces of the nonsingular affinor on these vectors are in the same linear relationship.
The affinor is well defined by its partial traces on the basis vectors, and it will be nonsingular if its traces form a basis.
The affinor product of two affinors will be special if and only if at least one of these affinors is special.

Literature

Norden A.P. Spaces of affine connection. - M .: Nauka, 1976 .-- S. 432.

Affinor

Properties

Literature

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