Definitions

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Definitions in Group theory:

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A tensor is an element in a direct product vector space.

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A spinor is a vector in C2, a complex, two-dimensional vector space.

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The tensor product space

is said to be a carrier space for an rth-order tensor product representation

of G

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A metric function on a vector space V is a mapping of a pair of vectors into a number in the field F associated with the vector space.

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Groups preserving bilinear symmetric metrics are called orthogonal.

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Groups preserving bilinear antisymmetric metrics are called symplectic

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Groups preserving sesquilinear symmetric metrics are called unitary.

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A manifold,in general, is a topological space which is homeomorphic to R^m locally.