**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

.

.

.

.

.

.

Kähler form is

.

The upper measuring is Kähler measuring.

.

.

**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 .

**4.**

References are below.

<On distance of word>Distance Theory /Tokyo May 5, 2004

Distance of Word / Complex Manifold Deformation Theory / Tokyo November 30, 2008

<On manifold>

From Cell to Manifold/ Cell Theory / June 2, 2007

[ Cell Theory / Continuation of Quantum Theory for Language / For LEIBNIZ and JAKOBSON ]

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