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of any codimension the notion of type number is introduced. Under the assumption that the type number is greater than 1 an equivalence theorem is proved.

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(1)

n

n

n

n+p

n

n+p

n

n

f (x)

n+p

x

n

x

n

X

X

[211]

(2)

X

n

X

n

X

x

n

x

x

X

n

n

x

n

x

n

x

n

n

X

ξ

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ξ

n

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ξ

n

x

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n

x

x

x

n

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Nx

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n

x

x

X

X

x

X

Ox

x

x

1

k

i

i

i

1

k

1

r

i

j

(3)

n

n+p

n

1

k

x

k

i=1

i

i

x

n

1

k

x

n

x

n

n

x

1

k

x

1

k

1

k

j

i6=j

i

i

k

i=1

i

i

i6=j

i

i

i6=j

i

i

j

i6=j

i

i

i

j

x

i

i

j

x

1

k

x

x

1

k

1

k

x

x

x

x

1

k

k+1

p

x

j

p

i=1

ij

i

j

ij

j

x

n

n

X

k

j=1

j

j

X

k

i=1

i

i

(4)

k

j=1

j

j

X

p

i=1 k

j=1

ij

j

i

X

k

i=1

i

i

x

n

i

k

j=1

ij

j

k

j=1

ij

j

ij

ij

i,j=1,...,k

i

k

j=

ij

j

j

s

j

s

1

k

i

k

i=1

i

i

i

i

i

k

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i

i

i

i

i

i

1

k

i

i

i=1,...,k

i

i

k

i=1

i

i

i

(5)

n+p

n

n+p

ξ

x

n

1

r

x

n

ξi

j

1

k

1

r

x

n

1

k

i

ji

i

i

i,j

ji

i

j

x

n

i,j

ji

i

j

x

n

i,j

ji

ξi

j

i,j

ji

ξi

j

ξi

j

i

j

n

n+p

n

n

n+p

h(Y,Z)

h(X,Z)

ξ

ξ

X

X

X

(6)

n

n

n

n+p

1

1

x

n

1

n

n+p

1

n+p

n

n+p

x

n

x

n

n

n+p

n

x

x

x

x

x

n

X

x

n

1

k

x

k+1

p

x

x

1

k

x

k

i=1

i

i

X

X1

X2

1X

y

2X

y

x

X

2X

k

i=1

i

i

n

x

k

i=1

i

X2

i

k

i=1

i

Y2

i

(7)

2X

i

X

i

x

X

i

k

i=1

i

X

i

k

i=1

i

i

x

n

n+p

n

ξ

F (ξ)

X

X

n

n+p

n

n+p

x

n

n

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1

k

x

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i

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i

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h(Y,Z)

h(X,Z)

¯h(Y,Z)

h(X,Z)¯

(8)

i

ξi

ξ¯i

i

ξi

ξ¯i

ξi

ξ¯i

ξ

F (ξ)

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X

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X

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Y

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n

x

x

n

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0

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γ

0

0

0

(0)).

(9)

[6] M. S p i v a k, A Copmprehensive Introduction to Differential Geometry , Vol. 5, Publish or Perish, 1979, 361–369.

INSTITUTE OF MATHEMATICS PEDAGOGICAL UNIVERSITY W. POLA 2

35-959 RZESZ ´OW, POLAND

R´evis´e le 8.4.1994

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