On the geometry of projective immersions - Project Euclid

J. Math. Soc. Japan

Vol. 41, No. 4, 1989

On the geometry of projective immersions By Katsumi NOMIZU* and Ulrich PINKALL (Received July 11, 1988)

In two preceeding papers [5] and [6] we have given a new general approach to classical affine differential geometry and established the basic results concerning the geometry of affine immersions. The purpose of the present paper is to begin the study of projective immersions. We shall concentrate our attention to the case of codimension one. In Section 1 we recall the notion of projective structure on a manifold and state relevant facts. In Section 2 we define the notion of projective and equiprojective immersions and related concepts such as totally geodesic and umbilical immersions. In Section 3 we study equiprojective immersions of a flat projective structure $(M, P)$ into a flat projective structure of one higher dimension and show that they are umbilical, provided dim $M\geqq 3$ and the rank of . We derive certain corollaries and determine all connected, compact, umbilical hypersurfaces in $RP^{n+1}$ . In Section 4 we prove the projective version of the theorem of Berwald which characterizes quadrics in affine differential geometry. In Section 5 we study the effect of a projective change of the ambiant connection on a nondegenerate hypersurface $M$, namely, how the affine normal, the Blaschke induced connection, the affine metric, and the cubic form change. We find that the difference tensor between the Blaschke connection and the LeviCivita connection is a projective invariant. We hope to find some more applications of these formulas in the study of nondegenerate hypersurfaces in $RP^{n+1}$ . $(\tilde{M},\tilde{P})$

$h\geqq 2$

1. Projective structure. We recall from [4] the notion of projective structure on a differentiable , manifold $M$. It is defined by an atlas of local affine connections where is an open covering of $M$ and is a torsion-free affine connection on such that in any nonempty intersection the connections projectively equivalent. Here, and are in general, two affine connections and V are said to be projectively equivalent if there is a l-form such that $P$

$(U_{a}, \nabla_{\alpha})$

$\{U_{a}\}$

$U_{\alpha}$

$V_{a}$

$\nabla_{\alpha}$

$U_{\alpha}\cap U_{\beta}$

$\nabla$

$\nabla_{\beta}$

$\mu$

*The first author was partially supported by an NSF grant. This paper was written while he was at ${\rm Max}- Planck$ -Institut f\"ur Mathematik, Bonn. He would also like to thank TU, Berlin, for their hospitality during his visit.

K. NOMIZU and U. PINKALL

608 (1)

$V_{X}Y=\nabla_{X}Y+\mu(X)Y+\mu(Y)X$

for any vector fields

$X$

and

$Y$

.

As usual, when $(M, P)$ is a projective structure, we consider a maximal atlas of local affine connections and write $(U, \nabla)\in(M, P)$ to mean that an affine connection on an open subset $U$ of $M$ belongs to the maximal atlas for the projective structure $(M, P)$ . In dealing with projective equivalence and projective structures we normally assume that each affine connection involved is locally equiaffine relative to a certain volume element; this condition is equivalent to the property that the Ricci tensor is symmetric. When two such equiaffine connections are projectively equivalent, it follows that $d\mu=0$ in (1) and that they have the same projective curvature tensor: $\nabla$

$W(X, Y)Z=R(X, Y)Z-[\gamma(Y, Z)X-\gamma(X, Z)Y]$ ,

(2)

is the dimenwhere 7 denotes the normalized Ricci tensor $Ric/(n-1)$ , where sion of the manifold. We also remark that if is an equiaffine connection and is a closed l-form and thus exact on a neighborhood $U$ , then the projective gives rise to an equiaffine connection on $U$ . It is known change of by (cf. Proposition 4 in [4]) that for a projective structure $(M, P)$ and for any volume element on $M$ there is a unique globally defined compatible with $P$ such that . Let us also recall that if dim $M\geqq 3$ , vanishing of the projective curvature tensor $W$ is a necessary and sufficient condition for to be projectively flat ( $i.e$ . projectively equivalent to a flat affine connection). If dim $M=2$ , then $n$

$\nabla$

$\mu$

$\nabla$

$\mu$

$\nabla$

$\omega$

$\nabla\omega=0$

$\nabla$

(3)

$(\nabla_{X}\gamma)(Y, Z)=(\nabla_{Y}\gamma)(X, Z)$

is a necessary and sufficient condition for projective flatness. We may define the notion of path for a projective structure $(M, P)$ . By a path we mean a curve in $M$ which, around each of its points, is a pregeo; desic relative to some $\nabla\in(M, P)$ , that is, for some function in this case, is a pregeodesic relative to every $\nabla\in(M, P)$ . $x_{t}$

$\nabla_{t}x_{t}=\rho(t)x_{t}$

$\rho(t)$

$x_{t}$

2. Projective immersion. Let $(M, P)$ and be differentiable manifolds each with a projective and structure defined by means of an atlas of local affine connections $n+P=\dim\tilde{M}$ $n=\dim M$ , respectively. We set . and projective An immersion is called a immersion if the following condition is satisfied: $(\tilde{M},\tilde{P})$

$(U_{\alpha}, \nabla_{\alpha})$

$(\hat{U}_{\beta},\tilde{\nabla}_{\beta})$

$f:Marrow\tilde{M}$

(A)

For each point

$x_{0}$

of

$M$,

there exist local affine connections

$(U, \nabla)\in$

Projective immersions

and

$(M, P)$

$(\tilde{U},\tilde{\nabla})\in(\tilde{M},\tilde{P})$

, where

$U$

and

609

are neighborhoods of is an affine immersion.

respectively, such that This means that there is a field of transversal subspaces that for any vector fields $X$ and $Y$ on $U$ we have $f:(U, \nabla)arrow(C,\tilde{\nabla})$

(4)

and

$\hat{U}$

$\tilde{\nabla}_{X}(f_{*}(Y))=f_{*}(\nabla_{X}Y)+\alpha(X, Y)$

See [6]. In the case where codimension field on $U$ such that

where

,

$p=1$ ,

$x_{0}$

$xarrow N_{x}$

$\alpha(X, Y)\in N$

on

$f(x_{0})$

$U$

,

such

.

there is a transversal vector

$\xi$

(5)

$\tilde{\nabla}_{X}(f_{*}(Y))=f_{*}(\nabla_{X}Y)+h(X, Y)\xi$

.

and be manifolds with affine connections. An affine imLet , where $P$ is a projective immersion mersion and are the projective structures determined by and , respectively. and or When condition (A) is satisfied, we can, in fact, pick which satisfies the condition. More precisely, we have or find $(\tilde{M},\tilde{\nabla})$

$(M, \nabla)$

$(M, P)arrow(\tilde{M},\tilde{P})$

$f:(M, \nabla)arrow(\tilde{M},\tilde{\nabla})$

$\tilde{P}$

$\tilde{\nabla}$

$\nabla$

$(\tilde{U},\tilde{\nabla})$

$(U, \nabla)$

$(0,\tilde{\nabla})$

$(U, \nabla)$

is a projective immersion, then PROPOSITION 1. If , (B) for any Point $x_{0}\in M$ and for any local affine connection , , there exists a local affine connection is a neighborhood of where $U$ immerneighborhood is an , that such is a where of affine $f:(M, P)arrow(\tilde{M},\tilde{P})$

$(\hat{U}, \tilde{\nabla})\in(\tilde{M},\hat{P})$

$\hat{U}$

$(U, \nabla)$

$f(x_{0})$

$f;(U, \nabla)arrow(\hat{U},\tilde{\nabla})$

$x_{0}$

sion;

point $x_{0}\in M$ and for any local affine connection is a sufficiently small neighborhood of , there exzsts a local , such that , where is a neighborhood of is a neighborhood of $U$ such that affine immersion, where (C)

for any

$(U, \nabla)$

$\tilde{U}$

$f:(U_{1}, \nabla)arrow(\hat{U}, \tilde{\nabla})$

$f(x_{0})$

$U_{1}\subset U$

$U_{1}$

$U$

affine connection

$x_{0}$

$(0,\tilde{\nabla})$

, where

and

is an

$f(U_{1})\subset O$

.

be any affine and be as in (A). Let PROOF. Let , where we , where is a neighborhood of connection belonging to neighborhood . may assume . Choose a such that of on which gives projective equivalence of Then there exists a l-form and . Then $(U, \nabla)$

$(O_{1},\tilde{\nabla}_{1})$

$(C,\tilde{\nabla})$

$(\tilde{M},\tilde{P})$

$\hat{U}_{1}$

$\hat{U}_{1}\subset O$

$f(x_{0})$

$U_{1}\subset U$

$f(U_{1})\subset 0_{1}$

$x_{0}$

$\tilde{\nabla}$

$C_{1}$

$\tilde{\mu}$

$\tilde{\nabla}_{1}$

$\tilde{\nabla}_{1_{X}}(f_{*}Y)=\tilde{\nabla}_{X}(f_{*}Y)+p(f_{*}X)f_{*}Y+p(f_{*}Y)f_{*}X$

$=f_{*}(\nabla_{X}Y+\mu(X)Y+\mu(Y)X)+\alpha(X, Y)$

,

where is a l-form on $M$ and $\alpha(X, Y)$ belongs to the transversal subspace $N$ for . Now we may pick the connection $(M, P)$ , where $\nabla_{1_{X}}Y=\nabla_{X}Y+\mu(X)Y+\mu(Y)X$. Then the equation above shows $\mu=f^{*}\tilde{\mu}$

$f:(U, \nabla)arrow(O_{1},\tilde{\nabla}_{1})$

that

$f:(U_{1}, \nabla_{1})arrow(\tilde{U}_{1},\tilde{\nabla}_{1})$

$(U_{1}, \nabla_{1})\in$

is an affine immersion.

be as in and The proof for (C) is similar. Let $(U_{1}, \nabla_{1})\in(M, P)$ , where , there is a closed l-form on $(U, \nabla)$

$U_{1}\subset U$

$(\tilde{U},\tilde{\nabla})$

$\mu$

(A). $U_{1}$

For any such that

610


for all vector fields $X$ and $Y$ . Now it is easy to . If we projectively change find a closed l-form on $U$ such that is an affine immersion. by using the form , then to

$\nabla_{1_{X}}Y=\nabla_{X}Y+\mu(X)Y+\mu(Y)X$

$\tilde{\nabla}$

$f^{*}\tilde{\mu}=\mu$

$\overline{\mu}$

$f:(U_{1}, \nabla_{1})arrow(\tilde{U},\tilde{\nabla}_{1})$


$\square$

$\tilde{\mu}$

REMARK. In stating conditions such as (A), (B) or (C), we shall from now on omit explicit mention of the domains of local affine connections. In many cases, it suffices to say that around a given point there is a local affine connection $\nabla\in(M, P)$ with such and such properties. A projective immersion $f:(M, P)arrow(\tilde{M},\tilde{P})$ is said to be totally geodesic at $x_{0}\in M$ if for any is totally geodesic relative to V at around is tangent , that is, for any vector fields $X$ and $Y$ around $Z$ . Now to $f(M)$ , that is, there is a vector at such , as is easily verified. this condition is independent of the choice of $h=0$ (5). equivalent in We say that is totally It is at to the condition that geodesic if it is so at every point of $M$. It is not difficult to see that is totally geodesic if and only if the image of a path in $M$ is a path for . From this point on, we shall deal with the case of codimension $p=1$ . Let $f:(M, P)arrow(\tilde{M},\tilde{P})$ be a projective immersion of codimension 1. For any point , there exist $\nabla\in(M, P)$ around and around such that is an affine immersion relative to and fi around . This means that there is a transversal vector field as in the equation (5). We show that the transat is independent of the pair (V, ) if is not totally versal direction geodesic at . be another pair we may choose. We can For this purpose, let and are defined in the same domain and projectively related: assume that $\nabla_{1_{X}}Y=\nabla_{X}Y+\mu(X)Y+\mu(Y)X$, where is a certain l-form, and, similarly for , where is a certain l-form. Suppose and : and is a transversal vector field for the affine immersion relative to $Z$ $M$ , where is a vector field tangent to and is a nonwrite vanishing function. From $\tilde{\nabla}\in(\tilde{M},\tilde{P})$

$f(x_{0}),$

$f$

$x_{0},$

$x_{0}$

$[\tilde{\nabla}_{X}(f_{*}(Y))]_{x_{0}}$

$thatf_{*}(Z)=[\tilde{\nabla}_{X}(f_{*}(Y))]_{x_{0}}$

$x_{0}$

$\tilde{\nabla}\in(\tilde{M},\tilde{P})$

$f$

$x_{0}$

$f$

$\tilde{M}$

$x_{t}$


$f(x_{0})$

$x_{0}$

$x_{0}$

$\nabla$

$f$

$x_{0}$

$\xi$

$\tilde{\nabla}$

$[\xi]$

$f$

$x_{0}$

$x_{0}$

$(\nabla_{1},\tilde{\nabla}_{1})$

$\nabla$

$\nabla_{1}$


$\mu$

$\tilde{\nabla}$

$\tilde{\nabla}_{1_{X}}Y=\tilde{\nabla}_{X}Y+\tilde{\mu}(X)Y+\tilde{\mu}(Y)X$

$\xi_{1}$

$\tilde{\mu}$

$f$

$\xi_{1}=f_{*}(Z)+\varphi\xi$

$\varphi$

$\tilde{\nabla}_{X}(f_{*}(Y))=f_{*}(\nabla_{X}Y)+h(X, Y)\xi$

and $\tilde{\nabla}_{1}x(f_{*}(Y))=f_{*}(\nabla_{1_{X}}Y)+h_{1}(X, Y)\xi_{1}$

we obtain (6)

$\nabla_{1}XY+h_{1}(X, Y)Z=\nabla_{X}Y+(f_{lJ}^{*\sim})(Y)X+(f_{\tilde{\mu}}^{*})(X)Y$

and (7)

$(\nabla_{1},\tilde{\nabla}_{1})$

$h(X, Y)=\varphi h_{1}(X, Y)$

.

611

Projective immersions

and take a at . Assume that $Z\neq 0$ at linearly independent from $Z$. We may take a geodesic tangent vector $X$ at and for with initial condition $(x_{0}, X)$ . This curve is a pregeodesic for $X$ is considered $X$ by a certain function , multiple where of of is a so as the tangent vector field of the curve . From (6) we obtain $h_{1}(X, X)Z=$ at , where is a scalar. Thus $h_{1}(X, X)=0$ . We have shown that for any $X\in T_{x_{0}}(M)$ linearly independent from $Z\neq 0$ , , be a basis of $T_{x_{0}}(M)$ . From we have $h_{1}(X, X)=0$ . Let $Z=X_{1},$ $h_{1}(X_{k}, X_{k})=0$ and $h_{1}(X_{j}, X_{k})=0$ what we proved, we have for each . We have also for all

Now we want to prove that

$Z=0$

$x_{0}$

$x_{0}$

$x_{0}$

$\nabla$

$\nabla_{1}$

$x_{t}$

$t$

$\nabla_{1_{X}}X$

$x_{t}$

$\lambda X$

$\lambda$

$x_{0}$

$X_{2},$

$k,$

$j,$

$X_{n}$

$\cdots$

$2\leqq k\leqq n,$

$k\geqq 2$

$h_{1}(X_{1}+X_{k}, X_{1}+X_{k})=h_{1}(X_{1}, X_{1})+2h_{1}(X_{1}, X_{k})=0$

as well as $h_{1}(X_{1}+2X_{k}, X_{1}+2X_{k})=h_{1}(X_{1}, X_{1})+4h_{1}(X_{1}, X_{k})=0$

which together imply that $h_{1}(X_{1}, X_{1})=h_{1}(X_{1}, X_{k})=0$ , where shown that $h_{1}=0$ at . This contradicts the assumption that $x_{0}$

,

$k\geqq 2$

$f$

.

We have

is not totally

.

geodesic at We may state this result as follows. $x_{0}$

be a pr0jective immerszon PROPOSITION 2. Let menston 1. There is a uniquely determined transversal direction field in the interior $V$ of the set of pmnts where is totally geodestc. $f:(M, P)arrow(\tilde{M},\tilde{P})$

for codiexcept

$[\xi]$

$f$

on $M-V$ the transversal direction field for the projective We shall call is determined up to a scalar immersion . The symmetric bilinear form $M$ and is at the points where is totally geodesic. We call the factor on $[h]$ the class conformal fundamental form for the projective immersion . The rank, uniquely determined at each point, is the rank of at the point. In particular, if the rank is is said to be nondegenerate at the point. We say that is nondegenerate if it is so at every point of $M$. Suppose that is not totally geodesic at and thus not in a neighborhood and a transversal field of . For any choice of relaimmersion, an affine is we write which to tive $[\xi]$

$h$

$f$

$0$

$f$

$f$

$f$

$f$

$n,$

$f$

$f$

$x_{0}$

$\nabla\in(M, P),\tilde{\nabla}\in(\tilde{M},\tilde{P})$

$\xi$

$x_{0}$

$f$

(8)

$I$

$x\xi=-f_{*}(SX)+\tau(X)\xi$

,

where is the shape operator for on $M$ and is the transversal connection , where changes to form for . If we change to is a function, then . Thus the condition that is a scalar multiple of the into and identity: does not change. Nor does the 2-form . may We also change to a projectively equivalent , where is a certain exact l-form. Then changes into $S$

$\xi$

$\xi$

$\xi$

$\tau$

$S$

$\varphi\xi$

$\varphi$

$\varphi S$

$\tau$

$S$

$\tau+d\varphi$

$S=\lambda I$

$d\tau$

$\tilde{\nabla}$

$\tilde{\nabla}_{X}Y+\tilde{\mu}(X)Y+\tilde{\mu}(Y)X$

$\tilde{\nabla}_{1}\in(\tilde{M},\tilde{P}):\tilde{\nabla}_{1}Y=X$

$\tilde{\mu}$

$S$

612


does not change. . Thus the condition and into Nor does the form . In view of this observation, we can make the following definition. A projective immersion , which is not totally geodesic at a point , is said to be if, for some choice of and relative to which is an affine umbilical at immersion, is a scalar multiple of the identity at . If is umbilical at every point, we say that is umbilical. We now introduce the notion of equiprojective immersion. Let $(M, P)$ and be two manifolds with projective structures and dim $\tilde{M}=\dim M+1$ . An is said to be equiprojective if immersion $S_{1}=S-\mu(\xi)I$

$S=\lambda I$

$\tau_{1}=\tau+f^{*}\tilde{\mu}$

$\tau$

$d\tau$

$f$

$x_{0}$

$\nabla,\tilde{\nabla}$

$f$

$\xi$

$x_{0}$

$S$

$f$

$x_{0}$

$f$

$(\tilde{M},\tilde{P})$

$f:Marrow\tilde{M}$

(A)

and

of $M$, there exist local equiaffine connections for each point such that is an affine immersion. $x_{0}$


$\nabla\in(M, P)$

$f$

and can be always chosen to be equiaffine. Just like Note that one of the case of mutually equivalent conditions (A), (B), (C) for projective immersions, we may state the following. $\tilde{\nabla}$

$\nabla$

is an $equipro_{J}$ ective immersion, then PROPOSITION 3. If $x_{0}\in M$ (B) for any point and for any local equiaffine connection around $\nabla\in(M, P)$ around such that is there exists a local equiaffine connection $f:(M, P)arrow(\tilde{M},\tilde{P})$


$f(x_{0})$

$f$

$x_{0}$

an affine immersion; (C) $x_{0}$

for any Point

, there

an affine

$x_{0}\in M$

for any local equiaffine connec tion

and

exists a local equiaffine connection immersion.

$\tilde{\nabla}\in(\tilde{M},\hat{P})$

around

around such that is

$\nabla\in(M, P)$

$f(x_{0})$

$f$

is an equiprojective immersion, then locally we may choose a When transversal vector field to be equiaffine relative to and (cf. Proposition 3 in [5]). For such a choice of , we have $\tau=0$ . Thus for an equiprojective immersion, we have $d\tau=0$ . We now state $f$

$\tilde{\nabla}$

$\nabla$

$\xi$

$\xi$

PROPOSITION 4. A pojective immerston if and only if $d\tau=0$ .

$f:(M, P)arrow(\tilde{M},\tilde{P})$

is equiprojective

PROOF. We prove that $d\tau=0$ implies that is equiprojective. Let and choose an equiaffine with a local parallel volume element $\nabla\in(M, P)$ around and an arbitrary around so that is an affine immer(with ). a transversal vector field Since $d\tau=0$ , there exists a function sion such that . Then for around we get the same connection

$x_{0}\in M$

$f$


$\tilde{\omega}$

$f(x_{0})$

$f$

$x_{0}$

$\xi$

$\varphi$

$x_{0}$

$\tau=-d\varphi$

$\overline{\xi}=\varphi\xi$

$\nabla$

induced, namely, $\tilde{\nabla}_{X}(f_{*}Y)=f_{*}(\nabla_{X}Y)+h(X, Y)\overline{\xi}$

and, on the other hand,

defined by

$\overline{\tau}=0$

.

,

This means that the local volume element

$\omega$

613

Projective immersions $\omega(X_{1}, \cdots , X_{n})=\tilde{\omega}(X_{1}, \cdots , X_{n},\overline{\xi})$

is parallel relative to to be equiprojective.

$\nabla$

(cf. [3]).

Thus

where

, $\nabla$

$X_{1},$

$\cdots$

,

is equiaffine, and

$X_{n}\in T_{x}(M)$

$f$

has been shown

$\square$

REMARK. If $d\tau=0$ , then there is a choice of for the affine immersion so that the equation $h(SX, Y)=h(X, SY)$ holds (see [5]). In the terminology of projective differential geometry, this property is expressed by saying that the normal congruence is conjugate (for instance, see [1], p. 31).

$f$

$\xi$

$\xi$

3. Equiprojective immersions between flat projective structures.

Now we recall (see [4]) that a projective structure $(M, P)$ is said to be flat if each local affine connection $\nabla\in(M, P)$ is projectively flat; in other words, if the atlas $(M, P)$ contains a flat affine connection around each point. We now prove be an equiprOjectjve immersion, where THEOREM 5. Let $\tilde{M}=n+1$ $M=n\geqq 3$ , dim . Assume that is flat. Then $(M, P)$ is flat if dim $p\alpha ntx_{0}\in M$ we have urther and only if at each 1) $S=\rho I$, or 2) rank $h=1$ and $S=\rho I$ on Ker , $f:(M, P)arrow(\tilde{M},\tilde{P})$

$(\tilde{M},\hat{P})$

$h$

$h=0$ .

3)

, choose equiaffine connecPROOF. Assume that $(M, P)$ is flat. For $\nabla\in(M, P)$ such that is an affine immersion with an equiand tions affine transversal vector field . is projectively flat, we have Since $x_{0}\in M$


$f$

$\xi$

$\tilde{\nabla}$

(9)

where (10)

$\tilde{R}(X, Y)Z=\tilde{\gamma}(Y, Z)X-\tilde{\gamma}(X, Z)Y$

$\tilde{\gamma}$

is the normalized Ricci tensor. Thus the Gauss equation

$\gamma(Y, Z)=\tilde{\gamma}(Y, Z)+$

Since we assume that

$\nabla$

[ $h(Y,$

$Z)$

$\gamma$

of

tr $S-h(SY,$

$\nabla$

$Z)$

says

.

is given by

] $/(n-1)$

.

is also projectively flat, we have an equation similar

to (9): (12)

(see [5])

$R(X, Y)Z=\tilde{\gamma}(Y, Z)X-\tilde{\gamma}(X, Z)Y+h(Y, Z)SX-h(X, Z)SY$

From this we find that the normalized Ricci tensor (11)

,

$R(X, Y)Z=\gamma(Y, Z)X-\gamma(X, Z)Y$

Using (11) in (12) and comparing it with (10) we find

.

614


(13)

$(n-1)[h(Y, Z)SX-h(X, Z)SY]$

.

$=[(trS)h(Y, Z)-h(SY, Z)]X-[(trS)h(X, Z)-h(SX, Z)]Y$

is projectively It is easy to see that, conversely, this equation implies that flat. , , , and $we’ 11$ show that $S=\rho I$. Let { Now assume that rank , of Ker $n-r$ such that the last basis a vectors form } be a basis , for . and the first vectors are orthonormal: $Y\neq Z$ $X\neq Y,$ from . If $r=n(\geqq 3)$ , choose Let be from $X$ (13) we get . From from . If $r