8/14/2019 Klein .

1/21

1 Klein

- Klein

, Klein (~1870)

, . , , Hilbert , .

, , :

1. .2. (

. ) R n.

. ( . . ) Minkowski .

KLEIN

() } 1"-"1f XX:f /f { .

, ()

.

:

8/14/2019 Klein .

2/21

2 Klein

f () f -1 , 1-1

f f -1=id x idx(x)=x Xx 1-1

.

,

f , g 1-1 , f g 1-1

f g . idx =f Xf f idf x = () , (G, ).

Klein:

(G, ) , Klein

, (G, ) (() , ).

Klein;

Klein .

(:=

)

G .

G;

G , f( )

f G.

:

8/14/2019 Klein .

3/21

3 Klein

J (:= )

( P( X) )

(J)={A P( X) : J(A) }

J f : X X

f(A) (J).

: J G

, f G.

G .

,

:

:

,

.

, ( u ,

f ,

u =: f(u) )

.

8/14/2019 Klein .

4/21

8/14/2019 Klein .

5/21

8/14/2019 Klein .

6/21

6 Klein

. .. .

, ,

.

. .

,

,

.

. :

, .

, :

,

. ,

.

,

, .

, dE

, ,

8/14/2019 Klein .

7/21

7 Klein

d

2 , )( E d ,

:

2 2( ), )d

:

2

2

1 ,2 f :

. f(1)=2

.

8/14/2019 Klein .

8/21

8 Klein

, 1 2

, g 1

2

. g : g( 1)= 2

~ 1~ 2

.

: ~

:

:

1 ()= ,

1 : .

1 G G . . : S(X) .

1~ 2 . 1 2: ( ) g g = . g

1-1 , , 1 1 1

1 2

11 2

11 2

2 1

( ) ( )

1 ( ) ( )

( )

~

g g g g

g

g

=

=

=

8/14/2019 Klein .

9/21

9 Klein

g-1 G , G

( S(X) )

~ ,

1 2 2 3

1 2 1 1 2

2 2 3 2 1 1

2 1 1 2 2 3

1 3

~ ~

, : ( )

( ) . , ( )

( ( )) ( )

~

g g G g

g T g g

g g g

=

= = = =

2 1 g g G g1 g2

G

.

,

G , .

;

,

g

.

R n n >3 .

R n n

.

8/14/2019 Klein .

10/21

10 Klein

R n .

;

,

,

:

f: V V ,

[ , ] V , ( [ , ]=[f(x),f( )] , V)

:

:F(x+ )=f( )+f( ) f( )= f( )

,

: [f( ) f(x) , f( )- f( )]=0 . (1)

,

(1) f( )- f( )=0 f( )= f( ).

, :

[f( ) f(x) , f( )- f( )]=

[ f( ) , f( )- f( )]-[ f(x) , f( )- f( )]=

[f( ) , f( )]-[ f( ) , f( )]-[ f(x) , f( )]+[ f(x) , f( )]=

[f( ) , f( )]- [ f( ) , f( )]- [ f(x) , f( )]+ 2[ f(x) , f( )] =

[ , ]- [ f( ) , f( )]- [f( ) , f(x)]+ 2[ f(x) , f( )] =

8/14/2019 Klein .

11/21

8/14/2019 Klein .

12/21

12 Klein

[ , ]+[ ,]+[, ]+[,]-[ , ]-[, ]-[ ,]-[,]-[,]-[ , ]+

[x ,x]+[ x, ] +[ ,x]+[ ,]=0

.---

[,] f f:X X ,

:

( ) f

( ) f f(0)=0

:

( ) ( ) > &

:

f(0+0)=f(0)+f(0)

f(0)=f(0)+f(0)

f(0)=0

( ) ( ) > f ,

.

8/14/2019 Klein .

13/21

13 Klein

,

, !

, , f

.

:

( ) ( ) , f x f y x y x y X = (1)

=0 (1) :

( ) (0) 0 ,0 f x f x x X = ( f(0)=0 )

( ) 0 0 ,0 f x x x X =

( ) , f x x x X =

f .

, y X :

8/14/2019 Klein .

14/21

14 Klein

2

2

2 2

2 2

2 2

[ ( ) ( ), ( ) ( )] ( ) ( )

[ , ][ , ] 2[ , ] [ , ]

2[ , ] ( )

( ) 2[ , ] ( )

:

[ ( ) ( ), ( ) ( )] ( ) 2[ , ] ( )

f x f y f x f y f x f y

x y

x y x y x x x y y y

x x y y f

f x x y f y

f x f y f x f y f x x y f y

= =

=

= + =

+ =

+

= +(*)

:

2 2

[ ( ) ( ), ( ) ( )] [ ( ), ( )] 2[ ( ), ( )] [ ( ), ( )]

( ) 2[ ( ), ( )] ( ) (**)

f x f y f x f y f x f x f x f y f y f y

f x f x f y f x

= + =

+

(*) (**) :

[f(x),f(y)]=[x,y] f ,

.---

8/14/2019 Klein .

15/21

8/14/2019 Klein .

16/21

16 Klein

( ) ( ) ( ( ) (0)) ( ( ) (0))( ) ( )

g x g y f x f f y f f x f y x y

= + + = =

g .

, g .

. f : = f(0).

g

f , :

( f g)(x) = f (g(x))= g(x)+f(0)= f(x) x V

g

f

:

, g1 f b

:

g1 f b =f

:f(x)=f(x)

(g1 f b)( ) =( f g)(x)

g1( )+b =g(x)+a (1)

8/14/2019 Klein .

17/21

17 Klein

g g1 ,

g1(0)=0= g(0) (1) :

0+=0+b a=b. g1( )= g(x) V.

.

,

, nxn

g

V. dimV=n :

1 11 1 1 11 1 1

1

1 1

n n n

n m mn n m mn n

x a a x a x a x

g x

x a a x a a x

= =

: f: V V

, ( ) x f x x V =

: f , :

0 ( ) (0) x f x f x V = f , f(0)=0

8/14/2019 Klein .

18/21

18 Klein

0 ( ) 0 x f x x V =

( ) x f x x V =

: :

( ) x f x x V = , x-y

:

( ) ( )

( ) ( ) ( )

f x y x y f

f x f y x y f

=

=

g : R 2 R 2

(2 2) :

( , ) ( , )

x a ax y g

y y

g x y ax y y

+ = = +

= + +

8/14/2019 Klein .

19/21

8/14/2019 Klein .

20/21

20 Klein

2 2 1

x a b ax y b f

y y c y c

x a b ax y b f

y y c y c

a

+ = + = + + + = + = + + + = = = b=c=0 .

:

x f

y y

=

.

=1 , =0 ,

f(x,y)=(x,y)+(b,c)

=1 , =0 b=c=0

f(x,y)=(x,-y).

,

, .

, f(x ,y i)=g(x ,y i) i=1,2,3 , f=g.

8/14/2019 Klein .

21/21

21 Klein

, , b, c

.

, ,

.

.-