Grassmann Language

Grassmann Language
 
 
 
 
1. is (N+1) dimensional complex space.
 is Grassmann manifold that is all the spaces which are made from (k+1) dimensional linear subspace of  .
Element of (k+1) dimensional exterior space of  is called (k+1) vector.
Elements of  is given by (k+1) vector of decomposable is givenby the below.
 .
 is one fixed base of .
 is also able to written.
 is called Cayley-Plücker-Grassmann coordinates.
 is unitary group.
 .
 .
Hermitian inner product is defined as the below.
 .
 .
 is called norm of  .
Over 
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.
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 .
 .
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Kähler form is
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The upper measuring is Kähler measuring.
 .
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2.(Theorem)
Grassmann manifold has invariant Kähler measuring by operation of  .
Its Kähler form is equal to  times of hyperplane section bundle's curvature form over  that is determined by immersion of Cayley-Plücker-Grassmann coordinates and hermitian norm  .
3.Grassmann language model is supposed as the below.
Word is expressed by grassmann manifold.
Word has location and distance, that are expressed by Cayley-Plücker-Grassmann coordinates and hermitian norm  .