### POLONICI MATHEMATICI LX.3 (1995)

**An equivalence theorem for submanifolds** **of higher codimensions**

## by Pawe lWitowicz (Rzesz´ow)

**Abstract. For a submanifold of R**

^{n}### 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.

## Introduction. The paper deals with submanifolds of R

^{n}

## of codimension greater than one, equipped with affine connections. Problems concerning higher codimensions have been studied in Riemannian geometry since the 1930-ties, for example by Allendoerfer (see [1]). He formulated and proved a few theorems about the existence and equivalence of submanifolds iso- metrically immersed in Euclidean spaces of high dimension ([1], [6]). A key notion used in his papers was the type number of an immersion, which is a generalization of the rank of the second fundamental form.

## In affine geometry there are very few results dealing with submanifolds of any codimension (see [2], [4]).

## In this paper the type number of an immersion of any codimension is defined in the affine case. Using this notion we prove an equivalence theorem.

## More precisely, we show that two submanifolds of type number greater than one having the same affine connections and second fundamental forms are affinely equivalent.

## 1. The type number of an immersion. Let f : M

^{n}

## → R

^{n+p}

## be an immersion of an n-dimensional manifold M

^{n}

## into R

^{n+p}

## equipped with the usual flat affine connection e ∇. If N is a vector bundle over M

^{n}

## such that (1.1) ∀x ∈ M

^{n}

## T

_{f (x)}

## R

^{n+p}

## = f

_{∗}

## (T

x## M

^{n}

## ) ⊕ N

x## ,

## then for all X, Y ∈ X (M

^{n}

## ) we have a decomposition (1.2) ∇ e

_{X}

## f

∗## (Y ) = f

∗## (∇

X## Y ) + h(X, Y ),

*1991 Mathematics Subject Classification: Primary 53A13; Secondary 53C40.*

*Key words and phrases: submanifold, affine immersion, normal bundle.*

[211]

## Here ∇

X## Y is a local section of T M

^{n}

## and h(X, Y ) is a local section of N . Throughout the paper we write ∇

X## Y ∈ T M

^{n}

## and h(X, Y ) ∈ N to mean that (∇

X## Y )(x) ∈ T

x## M

^{n}

## and h

x## (X, Y ) ∈ N

x## for every x in the domain of

## ∇ e

_{X}

## f

∗## (Y ). It is easy to prove that ∇ is an affine connection without torsion on M

^{n}

## and h, called the second fundamental form on M

^{n}

## , is a bilinear symmetric mapping from T

x## M

^{n}

## × T

x## M

^{n}

## into N

x## for each fixed x ∈ M

^{n}

## .

## For a local section ξ of N and for any X ∈ X (M

^{n}

## ) we also have a decomposition

## (1.3) ∇ e

_{X}

## ξ = −f

_{∗}

## (A

ξ## X) + D

X## ξ,

## where A

ξ## and D

X## ξ are defined by A

ξ## X ∈ T M and D

X## ξ ∈ N . A

ξ## is a (1,1)- tensor field on M

^{n}

## , called the shape operator , and D

X## ξ defines a connection in the vector bundle N , called the normal connection. We also remark that the mapping ξ 7→ A

ξ## is linear over C

^{∞}

## (M

^{n}

## ). For fixed N we define the subspace O

x## of N

x## by

## O

x## = span{h(X, Y ) | X, Y ∈ T

x## M

^{n}

## }, called the first affine normal space.

## Lemma 1.1. The dimension of O

x## does not depend on the choice of the transversal bundle N .

## P r o o f. Let N, N be two affine normal bundles, h, h the associated sec- ond fundamental forms and O

x## , O

x## the first affine normal spaces induced by h and h, respectively. For x ∈ M

^{n}

## let

## π

x## : f

∗## (T

x## M

^{n}

## ) ⊕ N

x## → N

x## be the projection. Then π

x## = π|

Nx## is an isomorphism. For every X, Y ∈ T

x## M

^{n}

## we have

## π

x## (h(X, Y )) = π

x## ( e ∇

X## f

_{∗}

## (Y ) − f

_{∗}

## (∇

X## Y )) = π

x## ( e ∇

X## f

_{∗}

## (Y )) = h(X, Y ).

## This implies that π|

Ox## is an isomorphism between O

x## and O

x## , which com- pletes the proof.

## We now adapt the notion of the type number of an immersion used in the Riemannian case to affine geometry. First we define the type number of a set of bilinear forms.

## Let V be an n-dimensional vector space and h

^{1}

## , . . . , h

^{k}

## be linearly in- dependent bilinear forms on V × V . We define mappings Φ

^{i}

## : V → V

^{∗}

## (for i = 1, . . . , k) as follows:

## Φ

^{i}

## (v)(w) = h

^{i}

## (v, w).

## The type number of the set {h

^{1}

## , . . . , h

^{k}

## } is the maximal integer r such that

## there exist r vectors v

1## , . . . , v

r## ∈ V for which Φ

^{i}

## (v

j## ) are linearly independent

## for i = 1, . . . , k and j = 1, . . . , r.

## Now we define the type number of an immersion f : M

^{n}

## → R

^{n+p}

## at x ∈ M

^{n}

## (compare [1], [3], [6]). Choose a basis {ξ

1## , . . . , ξ

k## } of O

_{x}

## . Then

## h(X, Y ) =

k

## X

i=1

## h

^{i}

## (X, Y )ξ

i## for every X, Y ∈ T

x## M

^{n}

## . It is clear that h

^{1}

## , . . . , h

^{k}

## : T

x## M

^{n}

## × T

x## M

^{n}

## → R are symmetric bilinear forms.

## Definition 1.2. The type number of an affine immersion f at x ∈ M

^{n}

## , denoted by t

x## f , is the type number of the set {h

^{1}

## , . . . , h

^{k}

## } of symmetric bilinear forms defined above.

## We prove that t

x## f is a well defined notion.

## Lemma 1.3. The forms h

^{1}

## , . . . , h

^{k}

## are linearly independent.

## P r o o f. Assume for contradiction that there exist numbers α

1## , . . . , α

k## such that

## h

^{j}

## = X

i6=j

## α

i## h

^{i}

## for some j ≤ k. Then

## h =

k

## X

i=1

## h

^{i}

## ξ

i## = X

i6=j

## h

^{i}

## ξ

i## + X

i6=j

## α

i## h

^{i}

## ξ

j## = X

i6=j

## h

^{i}

## (ξ

i## + α

i## ξ

j## )

## and now O

x## is spanned by the k − 1 vectors ξ

i## + α

i## ξ

j## for i ∈ {1, . . . , k}\{j}.

## Lemma 1.4. t

x## f is independent of the choice of N and {ξ

1## , . . . , ξ

k## }.

## P r o o f. Let N

x## and N

x## be two transversal spaces and let ξ

1## , . . . , ξ

k## and ξ

_{1}

## , . . . , ξ

_{k}

## span O

x## ⊂ N

_{x}

## and O

x## ⊂ N

_{x}

## respectively. Let ξ

_{1}

## , . . . , ξ

_{k}

## , ξ

_{k+1}

## , . . . , ξ

_{p}

## span the whole space N

x## . Then

## ξ

j## =

p

## X

i=1

## a

ij## ξ

_{i}

## + Z

j## ,

## where a

ij## ∈ R and Z

j## ∈ T

_{x}

## M

^{n}

## . If ∇ and ∇ are the connections on M

^{n}

## defined by N and N , then

## f

∗## (∇

X## Y ) +

k

## X

j=1

## h

^{j}

## (X, Y )ξ

j## = f

∗## (∇

X## Y ) +

k

## X

i=1

## h

^{i}

## (X, Y )ξ

_{i}

## and

k

## X

j=1

## h

^{j}

## (X, Y )Z

j## + f

∗## (∇

X## Y ) +

p

## X

i=1 k

## X

j=1

## a

ij## h

^{j}

## (X, Y )ξ

_{i}

## = f

∗## (∇

X## Y ) +

k

## X

i=1

## h

^{i}

## (X, Y )ξ

_{i}

## for every X, Y ∈ T

x## M

^{n}

## . Hence h

^{i}

## = P

kj=1

## a

ij## h

^{j}

## for i = 1, . . . , k and P

kj=1

## a

ij## h

^{j}

## = 0 for i = k + 1, . . . , p, which gives a

ij## = 0 for i > k by the previous lemma. Thus the matrix [a

ij## ]

i,j=1,...,k## is non-singular. Now the equation

## Φ

^{i}

## =

k

## X

j=

## a

ij## Φ

^{j}

## implies that if Φ

^{j}

## (X

s## ) are linearly independent for s = 1, . . . , r and j = 1, . . . , k then so are Φ

^{j}

## (X

s## ). The proof is complete.

## We now prove an algebraic lemma which will be useful later.

## Lemma 1.5. Let V ,W be vector spaces, and h

^{1}

## , . . . , h

^{k}

## be bilinear forms on V × V with type number greater than one. Let B

i## : V → W be linear maps for i = 1, . . . , k. If

## (∗)

k

## X

i=1

## {h

^{i}

## (X, Z)B

i## (Y ) − h

^{i}

## (Y, Z)B

i## (X)} = 0 for every X, Y, Z ∈ V , then B

i## = 0 for i = 1, . . . , k.

## P r o o f. It is sufficient to assume that W = R. The equation (∗) with fixed X and Y but arbitrary Z means that

## (∗∗)

k

## X

i=1

## {B

_{i}

## (Y )Φ

^{i}

## (X) − B

i## (X)Φ

^{i}

## (Y )} = 0,

## where Φ

^{i}

## : V → V

^{∗}

## is given by Φ

^{i}

## (v)(w) = h

^{i}

## (v, w). Because the type number of {h

^{1}

## , . . . , h

^{k}

## } is at least 2, we can take X, Y ∈ V such that the set {Φ

^{i}

## (X), Φ

^{i}

## (Y )}

i=1,...,k## is linearly independent. But this immediately implies B

i## (X) = B

i## (Y ) = 0. Now we substitute in (∗∗) an arbitrary vector Z ∈ V in place of Y and obtain

k

## X

i=1

## B

i## (Z)Φ

^{i}

## (X) = 0,

## whence B

i## (Z) = 0 for every Z ∈ V and i = 1, . . . , k. The proof is complete.

## R e m a r k 1.6. The definition of the type number of an immersion gener-

## alizes the corresponding definitions in the Riemannian case and in the case

## of hypersurfaces in affine geometry ([3], [6], [5]).

## Namely, if R

^{n+p}

## is equipped with the standard scalar product h , i and f : M

^{n}

## → R

^{n+p}

## is an immersion, we can take for N in (1.1) the normal bundle and for h in (1.2) the Riemannian second fundamental form with respect to h , i. Then hA

ξ## X, Y i = hh(X, Y ), ξi, where X, Y ∈ T

x## M

^{n}

## .

## According to [6], the type number of f is equal to the maximal integer r such that there exist r vectors X

1## , . . . , X

r## ∈ T

_{x}

## M

^{n}

## for which the set {A

ξi## X

j## : i = 1, . . . , k; j = 1, . . . , r} is linearly independent, where (ξ

1## , . . . , ξ

k## ) is a local frame on N . Let X

1## , . . . , X

r## ∈ T

_{x}

## M

^{n}

## , and let (ξ

1## , . . . , ξ

k## ) be a local frame on N . Let Φ

^{i}

## denote mappings as defined before Definition 1.2, and a

^{j}

_{i}

## real numbers. We have h

^{i}

## (X, Y ) = hh(X, Y ), ξ

i## i. Notice that the following conditions are equivalent:

## X

i,j

## a

^{j}

_{i}

## Φ

^{i}

## (X

j## ) = 0,

## ∀Y ∈ T

_{x}

## M

^{n}

## X

i,j

## a

^{j}

_{i}

## h

^{i}

## (X

j## , Y ) = 0,

## ∀Y ∈ T

_{x}

## M

^{n}

## X

i,j

## ha

^{j}

_{i}

## A

ξi## X

j## , Y i = 0, X

i,j

## a

^{j}

_{i}

## A

ξi## X

j## = 0.

## This means that the set {A

ξi## X

j## : i = 1, . . . , k; j = 1, . . . , r} is linearly independent if and only if {Φ

^{i}

## (X

j## ) : i = 1, . . . , k; j = 1, . . . , r} is. Thus the remark is true in the Riemannian case.

## In the case of hypersurfaces the type number of f is the rank of the second fundamental form, but the latter is equal to the rank of Φ.

## 2. Basic equations. We recall the equations of affine geometry (see for example [2]).

## Let f : M

^{n}

## → R

^{n+p}

## be an immersion. If we have N and an affine connection ∇ on M

^{n}

## such that (1.1) and (1.2) are satisfied, we call such an immersion an affine immersion and we write

## f : (M

^{n}

## , ∇) → (R

^{n+p}

## , e ∇).

## Denoting the curvature tensors of ∇ and D by R and R

^{⊥}

## respectively, we have the following equations:

## R(X, Y )Z = A

h(Y,Z)## X − A

h(X,Z)## Y (Gauss),

## ∇h(X, Y, Z) = ∇h(Y, X, Z) (Codazzi), R

^{⊥}

## (X, Y )ξ = h(X, A

ξ## Y ) − h(A

ξ## X, Y ) (Ricci),

## where ∇h(X, Y, Z) = D

X## h(Y, Z)−h(∇

X## Y, Z)−h(Y, ∇

X## Z) for every X, Y, Z

## ∈ X (M

^{n}

## ), and ξ ∈ X (N ). Here X (N ) denotes the module of local sections of the transversal bundle N over M

^{n}

## .

## 3. Reduction of codimension. We use a result from [4]:

## Proposition 3.1. Let f : (M

^{n}

## , ∇) → (R

^{n+p}

## , e ∇) be an affine immersion.

## Suppose that N

1## is a subbundle of the normal bundle N such that : (1) N

1## contains O

x## for every x ∈ M

^{n}

## .

## (2) N

1## is parallel relative to the normal connection D.

## Then f (M

^{n}

## ) is contained in an (n+q)-dimensional affine subspace of R

^{n+p}

## , where q = dim N

1## (x).

## We now formulate a reduction lemma which involves the notion of the type number of an affine immersion. It is actually a generalization of Lemma 28 from [6] to the case when the ambient manifold is the affine space (R

^{n+p}

## , e ∇).

## Proposition 3.2. Let f : (M

^{n}

## , ∇) → (R

^{n+p}

## , e ∇) be an affine immersion such that :

## (1) t

x## f ≥ 2 at every x ∈ M

^{n}

## . (2) dim O

x## = k on M

^{n}

## .

## Then f (M

^{n}

## ) is contained in an (n+q)-dimensional affine subspace of R

^{n+p}

## . P r o o f. We prove that the subbundle O of N satisfies the assumptions of the previous proposition. For every x ∈ M

^{n}

## let W

x## be a vector sub- space of N

x## such that N

x## = O

x## ⊕ W

_{x}

## . Let ξ be a section of O in an open neighbourhood of x ∈ M

^{n}

## . We now prove that (D

X## ξ)(x) ∈ O

x## for every X ∈ X (M

^{n}

## ).

## Let ξ

1## , . . . , ξ

k## be sections of N which span O

x## and ξ

k+1## , . . . , ξ

p## be sections which span W

x## . Then there exists a neighbourhood U

x## of x and functions a

1## , . . . , a

k## in U

x## such that ξ = P

ki=1

## a

i## ξ

i## . We also have a decomposition (D

X## ξ)(y) = (D

_{X}

^{1}

## ξ)(y) + (D

_{X}

^{2}

## ξ)(y),

## where (D

^{1}

_{X}

## ξ)(y) ∈ O

y## and (D

^{2}

_{X}

## ξ)(y) ∈ W

y## for y ∈ U

x## . The mapping X 7→ (D

X## ξ)(x) is obviously R-linear and so is X 7→ (D

^{2}X

## ξ)(x). Notice that the second fundamental form is h = P

ki=1

## h

^{i}

## ξ

i## and consider the Co- dazzi equation for arbitrary X, Y, Z ∈ X (M

^{n}

## ). Comparing its components belonging to W

x## gives

k

## X

i=1

## h

^{i}

## (Y, Z)D

_{X}

^{2}

## ξ

i## =

k

## X

i=1

## h

^{i}

## (X, Z)D

_{Y}

^{2}

## ξ

i## .

## By Lemma 1.5 we have (D

^{2}

_{X}

## ξ

i## )(x) = 0 and therefore (D

X## ξ

i## )(x) ∈ O

x## for i = 1, . . . , k. Hence (D

X## ξ

i## )(x) = P

ki=1

## a

i## (x)(D

X## ξ

i## )(x) + P

ki=1

## X(a

i## )ξ

i## (x)

## ∈ O

_{x}

## . An application of the above proposition completes the proof.

## 4. An equivalence theorem. Dillen’s paper [2] contains a general equivalence theorem for immersed manifolds of any codimension. Under an assumption about the type number we can formulate a stronger result. First we recall the result of Dillen.

## Theorem 4.1. Let f, f : (M

^{n}

## , ∇) → (R

^{n+p}

## , e ∇) be affine immersions with corresponding affine normal spaces N and N , second fundamental forms h and h, affine shape operators A and A, and normal connections D and D, respectively. Suppose that there exists an isomorphism F : N → N of vector bundles over M

^{n}

## such that :

## (1) F ◦ h = h.

## (2) A

ξ## = A

_{F (ξ)}

## .

## (3) F (D

X## ξ) = D

X## F (ξ), where X ∈ X (M

^{n}

## ) and ξ ∈ X (N ).

## Then there exists B ∈ A(n + p, R) such that f = B ◦ f , where A(n + p, R) denotes the group of affine transformations of R

^{n+p}

## .

## The main result of this section is the following theorem.

## Theorem 4.2. Let f, f : (M

^{n}

## , ∇) → (R

^{n+p}

## , e ∇) be affine immersions.

## Suppose that :

## (1) t

x## f ≥ 2 at every point x in M

^{n}

## .

## (2) There exists an isomorphism F : N → N of vector bundles over M

^{n}

## such that F ◦ h = h.

## (3) dim O

x## = k for every x ∈ M

^{n}

## .

## Then there exists B ∈ A(n + p, R) such that f = B ◦ f.

## P r o o f. Proposition 3.2 allows us to consider only the case of p = k. By (2), we have dim O

x## = dim O

x## . We take a set {ξ

1## , . . . , ξ

k## } of sections of N that span O

x## in a certain neighbourhood of a fixed point of M

^{n}

## .

## Let {ξ

_{1}

## , . . . , ξ

_{k}

## } be the sections given by F (ξ

_{i}

## ) = ξ

_{i}

## for i = 1, . . . , k. Then there exist two sets of symmetric bilinear forms on T

x## M

^{n}

## , {h

^{1}

## , . . . , h

^{k}

## } and {h

^{1}

## , . . . , h

^{k}

## }, such that h = P h

^{i}

## ξ

i## and h = P h

^{i}

## ξ

_{i}

## .

## By the assumptions we have X h

^{i}

## ξ

_{i}

## = F X

## h

^{i}

## ξ

i## = X

## h

^{i}

## ξ

_{i}

## . Hence h

^{i}

## = h

^{i}

## for i = 1, . . . , k.

## The Gauss equations for f and f imply that

## A

h(Y,Z)## (X) − A

h(X,Z)## (Y ) = A

¯h(Y,Z)## (X) − A

h(X,Z)¯## (Y ),

## whence

## X h

^{i}

## (Y, Z)(A

ξi## − A

ξ¯i## )(X) = X

## h

^{i}

## (X, Z)(A

ξi## − A

ξ¯i## )(Y ).

## Using Lemma 1.5 we obtain A

ξi## = A

ξ¯i## for i = 1, . . . , k. This also means that A

ξ## = A

F (ξ)## for every section ξ of N . The condition (2) from the previous theorem is satisfied.

## Now we apply F to the Codazzi equation for f and use the equality F ◦ h = h:

## 0 = F (∇h(X, Y, Z) − ∇h(Y, X, Z))

## = F (D

X## h(Y, Z) − D

X## h(X, Z))

## − (h(∇

_{X}

## Y, Z) + h(Y, ∇

X## Z) − h(∇

Y## X, Z) − h(X, ∇

Y## Z)).

## Next we compare the above equality with the Codazzi equation for f : 0 = D

X## h(Y, Z) − D

Y## h(X, Z)

## − (h(∇

X## Y, Z) + h(Y, ∇

X## Z) − h(∇

Y## X, Z) − h(X, ∇

Y## Z)) to obtain

## D

X## h(Y, Z) − D

Y## h(X, Z) = F D

X## h(Y, Z) − D

X## h(X, Z).

## Since h

^{i}

## = h

^{i}

## and F (ξ

i## ) = ξ

_{i}

## , by straightforward computation we get X h

^{i}

## (Y, Z)D

X## ξ

_{i}

## − F (D

_{X}

## ξ

i## ) = X

## h

^{i}

## (X, Z)D

Y## ξ

_{i}

## − F (D

_{Y}

## ξ

i## ).

## But the mappings X 7→ D

X## ξ

_{i}

## − F (D

_{X}

## ξ

i## ) are linear for i = 1, . . . , k. There- fore from Lemma 1.5 we have F (D

X## ξ

i## ) = D

X## ξ

_{i}

## for i = 1, . . . , k, which also implies F (D

X## ξ) = D

X## ξ for every section ξ of N . By Theorem 4.1, this completes the proof.

## R e m a r k 4.3. The affine transformation B obtained in Theorem 4.2 is unique (comp. [6]). Since O

x## = N

x## , it is enough to prove that each vector X ∈ O

x## is of the form (f ◦ γ)

^{00}

## (0) for a curve γ on M

^{n}

## . It is clear that the space O

x## is spanned by vectors of the form h(X, X), where X ∈ T

x## M

^{n}

## . If we take a geodesic γ on M

^{n}

## such that γ

^{0}

## (0) = X, then

## (f ◦ γ)

^{00}

## (0) = e ∇

_{γ}

## (f ◦ γ)

^{0}

## (0) = h(γ

^{0}

## (0), γ

^{0}

## (0)).

**References**

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