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J. Math.^ Soc.^ Japan Vol. 20,^ No. 3,^1968
Differential geometry^ of complex^ hypersurfaces II*
By Katsumi NOMIZU and Brian SMYTH
(Received (^) Jan. 8,^ 1968) In this paper (^) we continue the study of complex hypersurfaces (^) of complex
space forms ($i$. $e$^. K"ahlerian manifolds of constant holomorphic sectional curva-
ture) begun^ in [8]. The main results are: the determination of the holonomy
groups of such hypersurfaces, a generalization of the main theorem of [8] on
Einstein hypersurfaces,^ the non-existence of a certain type of hypersurface in
the complex^ projective^ space,^ and some results concerning^ the curvature of
complex (^) curves.
Let $\tilde{M}$ be a complex space form (which in general will not be complete)
of complex dimension $n+1$ and let $M$^ be an immersed complex hypersurface
in $\tilde{M}$. In \S 1 we show that the rank of the second fundamental form of $M$^ is
intrinsic and that $M$^ is rigid in $\tilde{M}$ , if the latter is simply connected and com-
plete. The local version of rigidity is contained as a special case in the work
of Calabi [1],^ but our method is more direct and more in the line of classical differential geometry.
The holonomy^ group^ of $M$^ (with^ respect to the induced K"ahler^ metric)^ is
studied in \S 2. If the holomorphic sectional curvature $\tilde{c}$ of $\tilde{M}$ is negative, the
holonomy group^ is always^ $U(n)$^. In the^ case^ where $\tilde{c}>0(e.^ g.\tilde{M}=P^{n+1}(C))$^ ,
the holonomy group^ of^ $M$^ is^ either^ $U(n)$^ or^ $SO(n)\times S^{1}(S^{1}$^ denotes the circle
group), the latter case arising only when $M$^ is locally holomorphically isometric
to the complex quadric $Q^{n}$^ in $P^{n+1}(C)$^. When $\tilde{c}=0$^ (i.^ e. when $\tilde{M}$ is flat),^ the
holonomy group of $M$^ depends on the rank of the second fundamental form
and we obtain a result of Kerbrat [3] more directly.
In \S 3 we first obtain the following^ generalized^ local version of the clas-
sification theorem of [8]. If the Ricci tensor $S$^ of $M$^ is parallel^ $(i.^ e.^ \nabla S=0)$^ ,
then $M$^ is totally^ geodesic^ in^ $\tilde{M}$ or else $\tilde{c}>0$^ and^ $M$^ is^ locally^ a^ complex
quadric. (^) To prove (^) this we modify Theorem 2 [8] to show that $M$^ is locally
symmetric when its Ricci tensor is parallel, and obtain the local classification
without using^ the^ list^ of^ irreducible^ Hermitian^ symmetric^ spaces.^ This^ local
version (^) was proved by Chern [2] with the original^ assumption^ that $M$^ is
Einstein, and Takahashi [9] has shown that $M$^ is Einstein if its Ricci tensor
*This work was partially supported by grants from the National Science Foundation.
Differential geometry^ of complex^ hypersurfaces^ II^499
is parallel. It is worth noting that when $c\sim\neq 0$^ this latter result follows im-
mediately from Theorem 2 of \S 2. We conclude this section with a better
global version of the classification theorem of $[8]$ –here the proof is made
considerably more elementary than the original one and simple-connectedness of (^) the hypersurfaces (^) is no longer (^) assumed in the case $\tilde{c}\leqq 0$.
We show,^ in^ \S 4, that^ the^ rank^ of^ the^ second fundamental form cannot be
identically equal to 2 on a compact complex hypersurface in $P^{n+1}(C),$^ $n\geqq 3$^.
In \S 5 we discuss the Gaussian mapping of a complex hypersurface $M$^ in $C^{n+1}$
into $P^{n}(C)$^ ;^ we find that its (^) Jacobian is essentially the second fundamental form and we show how the Gaussian mapping relates the K"ahlerian^ connec-
tions of $M$^ and $P^{n}(C)$^.
The study of complex curves in a 2-dimensional complex space (^) form is taken up in (^) \S 6. First (^) we take care of the case $n=1$^ in (^) Theorems 4 and 5.
We then obtain some characterizations of $P^{1}$^ and $Q^{1}$^ among closed nonsingular
complex curves in $P^{2}(C)$^ by curvature conditions. We shall use the same notation as in [8].
\S 1. Rigidity.
Let $M$^ be^ a^ K"ahler^ manifold^ of^ complex^ dimension^ $n$^ and^ let^ $f$^ be a
K"ahlerian (^) immersion ($i$. $e$^. a complex (^) isometric immersion) (^) of $M$^ as a complex
hypersurface in a space $\tilde{M}$ of constant holomorphic curvature $ c\sim$. For each
point $x_{0}\in M$^ there is a neighborhood $U(x_{0})$^ of $x_{0}$^ in $M$^ on which Gauss’ equa-
tion for the immersion $f$^ may^ be written as
$R(X, Y)=\tilde{R}(X, Y)+D(X, Y)$
with
$\tilde{R}(X, Y)=\frac{c\sim}{4}{X\wedge Y+JX\wedge JY+2g(X, JY)J}$ and $D(X, Y)=AX$ A $AY+JAX\wedge JAY$^ ,
where $X\wedge Y$^ denotes the skew-symmetric^ endomorphism^ which maps^ $Z$^ upon
$g(Y, Z)X-g(X, Z)Y$, and $X,$^ $Y,$^ $Z$^ are tangent vectors to $M$^ (see Proposition 3
[8]). Whereas $A$^ depends on the immersion $f$^ and on a local choice of unit
vector field^ normal^ to^ $M$,^ the following^ lemma^ shows that^ its^ kernel^ does not.
LEMMA 1. At each point^ $x\in U(x_{0})$^ we^ have
$KerA=$ { $X\in T_{x}(M)|D(X,$ $Y)=0$ for all $Y\in T_{x}(M)$ }
$=$ (^) { $X\in T_{x}(M)|(R-\tilde{R})(X,$^ $Y)=0$ (^) for all $Y\in T_{x}(M)$ (^) }.
PROOF. Clearly^ $Ker$^ $A$^ is contained^ in^ the^ subspace^ defined^ by^ $D$^.^ On^ the
other hand, if $X\not\in KerA$^ then $D(X, JX)=-2AX\wedge JAX\neq 0$, and the first
Differential geometry^ of complex^ hypersurfaces^ II^501
Now
$(R-\tilde{R})(e_{i}, Je_{i})=-2Ae_{i}\wedge JAe_{i}=-2\overline{A}e_{i}\wedge J\overline{A}e_{i}$ and the middle form of this identity being nonzero when $i\leqq k$^ ,^ we see that
$\overline{A}e_{i}$ is a linear combination of $Ae_{i}$ and $JAe_{i}$ , say
$\overline{A}e_{i}=\alpha_{i}Ae_{i}+\beta_{i}JAe_{i}$ .
It is then clear that $\alpha_{i}^{2}+\beta_{i}^{2}=1$^. From
$R(e_{i}, e_{j})-\tilde{R}(e_{i}, e_{j})=Ae_{i}\wedge Ae_{j}+JAe_{i}\Lambda JAe_{j}$ $=\overline{A}e_{i}\Lambda\overline{A}e_{j}+J\overline{A}e_{i}\Lambda J\overline{A}e_{j}$
we can easily deduce that $\alpha_{i}=\alpha_{j}=\alpha$ , say,^ and $\beta_{i}=\beta_{j}=\beta$^ ,^ say,^ for^ $1\leqq i,$^ $j\leqq k$^.
However $KerA=Ker\overline{A}$,^ by^ virtue of^ Lemma 1,^ and^ therefore^ $\overline{A}=\alpha A+\beta JA$
with $\alpha^{2}+\beta^{2}=1$^ at each point^ of a neighborhood^ of^ $x_{0}$^.^ By^ virtue of^ the
assumption on the rank of $M$^ at $x_{0}$^ we can (^) find a differentiable vector field $X$
on a neighborhood of $x_{0}$^ such that $AX\neq 0$^ ;^ and,^ since $\alpha=\frac{g(\overline{A}X,AX)}{g(AX,AX)}$^ ,^ it
follows that $\alpha$ (and similarly $\beta$ ) is a differentiable function on a neighborhood
of $x_{0}$^. We may^ then define a differentiable function $\theta$^ on^ a^ neighborhood^ $U(x_{0})$
of $x_{0}$^ such that $\alpha=\cos\theta$^ and $\beta=\sin\theta$^. Then $\xi^{\prime}=\cos\theta\xi+\sin\theta J\xi$^ is a unit
normal vector field on $U(x_{0})$^ with respect to the immersion $f$^ and clearly $A^{\prime}=\overline{A}$.
By Lemma 2,^ it follows that $s^{\prime}=\overline{s}$ also.
THEOREM 1.^ A^ connected^ Kahlerian hypersurface^ $M$^ of complex^ dimension
$n\geqq 1$ of a simply connected complete complex space form $\tilde{M}$ is rigid in $\tilde{M}$.
PROOF. If $R=\tilde{R}$^ at every point^ of $M$,^ then $M$^ has constant holomorphic
sectional curvature $ c\sim$. Therefore, by Corollary 2 of [8, \S 3], $M$^ is totally
geodesic (^) in $\tilde{M}$ and (^) thus is rigid. (^) If $R\neq\tilde{R}$ at some point (^) of $M$, (^) let $x_{0}$ be a
point where the rank of $M$^ is maximal. Let $f,\overline{f}:M\rightarrow\tilde{M}$ be two K"ahlerian
immersions. By virtue of Lemma 3,^ there exists a neighborhood $U(x_{0})$^ of $x_{0}$
and suitably chosen unit normal vector fields $\xi$ and $\overline{\xi}$ on $U(x_{0})$^ with respect
to the immersions $f$^ and $\overline{f}$^ respectively^ such that $A=\overline{A}$^ and $s=\overline{s}$^ on $U(x_{0})$^.
We now resort to local coordinates to show that $f$^ and $\overline{f}$ differ by a holomor-
phic motion $\phi$ of $\tilde{M}$ on $U(x_{0})$^ , that is, $\overline{f}=\phi\circ f$^ on $U(x_{0})$^ ; and, by analyticity,
this will then hold on all of $M$. In fact,^ since the group^ of holomorphic
isometries of $\tilde{M}$ is transitive on the set of unitary frames,^ we may assume
without loss of generality that
$f(x_{0})=\overline{f}(x_{0})$ (^) , $f_{}(x_{0})=\overline{f}_{}(x_{0})$ (^) , $\xi(x_{0})=\overline{\xi}(x_{0})$ (^) ,
where $f_{}$^ and $\overline{f}_{}$ denote the differentials of $f$^ and $\overline{f}$ , respectively, and prove
that $f=\overline{f}$^ in a^ neighborhood^ of^ $x_{0}$^.^ Let $(x^{1}, \cdots , x^{2n})$^ be a^ system of local co-
\langle ) $rdinates$ on $U(x_{0})$ and let $(u^{1}, \cdots , u^{2n+2})$ be a system of local coordinates on a
neighborhood of $f(x_{0})$^ in $\tilde{M}$ derived from a system of complex coordinates.
(^502) K. (^) NOMIZU and B. (^) SMYTH
We agree^ on the following^ ranges^ for the indices:
$1\leqq i,$ $j,$ $k,$^ $1\leqq 2n$ , $1\leqq p,$ $q,$ $r,$ $s\leqq 2n+2$ .
Our notation (in^ the summation convention)^ will be
$f^{p}(x)=u^{p}(f(x))$ (^) , $f^{p_{i}}=\frac{\partial f^{p}}{\partial x^{i}}$ , $f_{ij}^{p}=\frac{\partial^{2}f^{p}}{\partial x^{i}\partial x^{j}}$ , (^) etc. , $f_{*}(\frac{\partial}{\partial x^{i}})=f^{p_{i}}(\frac{\partial}{\partial u^{p}})$ ,
$\xi=\xi^{r}\frac{\partial}{\partial u^{r}}$ , $J\xi=(J\xi)^{r}\frac{\partial}{\partial u^{r}}$ , $\xi_{i^{r}}=\frac{\partial\xi^{r}}{\partial x^{i}}$ , $\xi_{ij}^{r}=\frac{\partial^{2}\xi^{r}}{\partial x^{i}\partial x^{j}}$ , etc..
The corresponding (^) notation for $\overline{f}$ is then self-explanatory. We also use $h_{ij}=h(\frac{\partial}{\partial x^{i}},$ $\frac{\partial}{\partial x^{j}})$ , $k_{ij}=k(\frac{\partial}{\partial x^{i}},$ $\frac{\partial}{\partial x^{j}})$ ,
$A\frac{\partial}{\partial x^{i}}=a^{j_{i}}\frac{\partial}{\partial x^{j}}$ , $s(\frac{\partial}{\partial x^{i}})=s_{i}$.
(Note that we have $A=\overline{A}$^ and $s=\overline{s}$^ so that we do not need the corresponding
notation for $\overline{f}$ here). The Christoffel symbols are denoted by $\Gamma_{jk}^{i}$ for $(x^{1}, \cdots , x^{2n})$
and by^ $\Gamma_{qr}^{p}$ for $(u^{1}, \cdots , u^{2n+2})$^. We note that $(J\xi)^{r}=-\xi^{r+n+1}$^ and $(J\xi)^{r+n+1}=\xi^{\gamma}$
(indices are understood here modulo $2n+2$) because of the nature of the
coordinate system $(u^{1}, \cdots , u^{2n+2})$^. The equations
$\tilde{\nabla}{f*(\frac{\partial}{\partial x^{i}})}f{}(\frac{\partial}{\partial x^{j}})=f_{}[\nabla_{-,\partial x}\partial_{\tau^{-}}\frac{\partial}{\partial x^{j}}]+h[\frac{\partial}{\partial x^{i}},$^ $\frac{\partial}{\partial x^{j}}]\xi+k[\frac{\partial}{\partial x^{i}},$^ $\frac{\partial}{\partial x^{j}}]J\xi$^ , $\tilde{\nabla}{f*(\frac{\theta}{\partial x^{i}})}\xi=-f{*}[A\frac{\partial}{\partial x^{i}}]+s[\frac{\partial}{\partial x^{i}}]J\xi$
for the immersion $f$^ then yield
(I) $f_{ij}^{r}=-f_{i}^{p}f_{j}^{q}\Gamma_{pq}^{r}+f_{k}^{r}\Gamma_{\dot{\tau}j}^{k}+h_{ij}\xi^{\gamma}+k_{ij}(J\xi)^{r}$^ , (II) $\xi_{i^{r}}=-f_{l}^{p}\xi^{q}\Gamma_{pq}^{r}-a_{i}^{j}f_{j^{r}}+s_{i}(J\xi)^{r}$^.
We denote the corresponding^ equations^ for the immersion $\overline{f}$^ by^ (I)^ and^ (II).
At $x_{0}$^ we have
(1) $f^{p}(x_{0})=\overline{f}^{p}(x_{0})$^ ,^ $f_{i}^{p}(x_{0})=\overline{f}{i}^{p}(x{0})$^ ,^ $\xi^{r}(x_{0})=\overline{\xi}^{r}(x_{0})$^ ,^ $(J\xi)^{r}(x_{0})=(J\overline{\xi})^{r}(x_{0})$^.
We wish to^ show that^ $f=\overline{f}$^ in^ a^ neighborhood^ of^ $x_{0}$^ ;^ since^ $f^{p}$^ and^ $\overline{f}^{p}$^ are^ real
analytic it suffices to prove
(2) $f_{ij}^{p}(x_{0})=\overline{f}{i}^{p{j}}(x_{0})$^ , (4) $f_{ijk}^{p}(x_{0})=\overline{f}{i}^{p{jk}}(x_{0})$^ , and so on for all higher-order^ derivatives at $x_{0}$^. (2)^ follows from^ (I),^ (I),^ (1)
and the equation^ $A=\overline{A}$^ on^ $U(x_{0})$^ ,^ while
(3) $\xi_{i^{r}}(x_{0})=\overline{\xi}{t^{r}}(x{0})$
follows from (II),^ (II), (1)^ and^ the equations^ $A=\overline{A}$^ and^ $s=\overline{s}$^ on^ $U(x_{0})$^.^ Now
504 K. NOMIZU and B. SMYTH $\mathfrak{h})$ of this matrix algebra. (^) For the sake of brevity we frequently use the same
symbol to denote an endomorphism of $T_{x_{0}}(M)$^ and its matrix with respect to
the above basis. We shall say that $M$^ is nondegenerate when $J\in \mathfrak{h}$^ and this
definition is independent of the point $x_{0}$^ (see [4], where the notion of non-
degeneracy (^) was defined to mean $J\in H$ ).
In this section all indices range from 1 to $n$^ and we agree that $i\neq j$^. Let
$E_{j}^{i}$ denote the $n\times n$ matrix whose $(i, j)$ entry (i-th row, j-th column) is 1 and
whose $(j, i)$^ entry is $-1$^ , all other entries being zero. For $p\neq q$^ as well as
$p=q$, let $F_{q^{1?}}$^ denote the $n\times n$^ matrix whose $(p, q)$^ and $(q, p)$^ entries equal 1,^ all
other entries being^ zero. Setting^ $K_{j}^{i}=[{0E^{0{j^{i}}}}^{E_{j}^{i}}]$^ and^ $S_{q^{p}}=[{F{q^{p}}0}^{0-F_{q}^{p}}]$^ (including $p=q)$ (^) , the following identities are readily verified (assuming $i\neq j$^ as agreed): \langle 9) $\left{\begin{array}{l}[K_{j}^{i},S_{k}^{i}]=-S_{k}^{j} (k\neq j),\[K_{j}^{i},S_{j}^{i}]=2(S_{i}^{i}-S_{j}^{j}),\[S_{j}^{i},S_{i}^{i}]=K_{j}^{i},\end{array}\right.$
where $[, ]$^ denotes the usual bracket operation.
The holonomy^ algebra^ $\mathfrak{h}$^ contains^ all^ curvature^ transformations^ of^ $T_{x_{0}}(M)$
and in particular the endomorphisms $R(e_{i}, e_{j}),$^ $R(e_{i}, Je_{j})$^ and $R(e_{i}, Je_{i})$^ for all
$i,$ $j$ . Their matrices with respect to the above basis are respectively
$(\lambda_{i}\lambda_{j}+\frac{\tilde{c}}{4})K_{j^{i}}$ , $-(\lambda_{i}\lambda_{j}-\frac{c\sim}{4})S_{j}^{i}$ (^) and $-\frac{c\sim}{2}J+2(\lambda_{i}^{2}-\frac{\tilde{c}}{4})S_{i}^{i}$ (^) ,
as may^ be^ verified by^ using^ (7).^ In^ the^ proofs^ which^ follow^ we^ make^ repeated
use of the^ fact^ that^ these^ are^ elements^ of^ $\mathfrak{h}$^.
LEMMA 4. Let $c\sim>0$^.
i) $K_{\iota^{k}}\in \mathfrak{h}$ for all $k,$^ $1(k\neq l)$^.
ii) If $S_{j}^{j}\in \mathfrak{h}$ for some $j$^ , then $\mathfrak{h}=n(n)$^.
iii) (^) If $S_{j^{i}}\in \mathfrak{h}$^ and $\lambda_{i}\neq\lambda_{j}$ for some pair $(i, j)$^ , then $\mathfrak{h}=\mathfrak{u}(n)$^.
PROOF. i)^ Since $\lambda_{k}\geqq 0$^ for^ all^ $k$^ and^ $\tilde{c}>0,$^ $R(e_{k}, e_{\iota})\in \mathfrak{h}$^ implies^ $K_{\iota^{k}}\in \mathfrak{h}$^ for
every pair $(k, l)$^.
ii) For $k\neq j$^ ,^ we have $[K_{k}^{j}, S_{j}^{j}]=-S_{k}^{j}\in \mathfrak{h}$^ using^ (i)^ and^ the^ assumption.
Thus $[K_{k}^{j}, S_{k}^{j}]=2(S_{j}^{j}-S_{k}^{k})\in \mathfrak{h}$^ and^ hence $S_{k}^{k}\in \mathfrak{h}$^ for^ all^ $k$^.^ In^ addition,^ $[K_{\iota^{k}}, S_{k}^{k}]$
$=-S_{\iota^{k}}\in \mathfrak{h}$ when $k\neq l$^. Since $K_{j^{i}}$ for all $i\neq j$^ and $S_{q}^{p}$^ for all $p,$ $q$ together span
$\iota\downarrow(n)$ , we have $\mathfrak{h}=u(n)$ (^). iii) By (i) and by^ the^ assumption,^ we have^ $[K_{j}^{i}, S_{j}^{i}]=2(S_{i}^{;}-S_{j}^{j})\in \mathfrak{h}$^.^ Since $R(e_{i}, Je_{i})-R(e_{j}, Je,)=-\frac{c\sim}{2}(S_{i}^{i}-S_{j}^{j})+2(\lambda_{i}^{2}S_{i}^{i}-\lambda_{j}^{2}S_{j}^{j})$ $=(2\lambda_{i}^{2}-\frac{\tilde{c}}{2})(S_{i}^{i}-S_{j}^{j})+2(\lambda_{i}^{2}-\text{\‘{A}}{j}^{2})S{j}^{j}$ belongs to $\mathfrak{h}$ , we infer that $(\lambda_{i}^{2}-\lambda_{j}^{2})S_{j}^{j}\in \mathfrak{h}$^ and^ hence^ $S_{j}^{j}\in \mathfrak{h}$^ since^ $\lambda_{i}\neq\lambda_{j}$^.^ By
Differential geometry^ of complex^ hypersurfaces^ II^505
(ii), we have $\mathfrak{h}=n(n)$^.
THEOREM 2.^ Let^ $M$^ be^ a^ complex^ hypersurface^ of complex^ dimension^ $n\geqq 1$ in a^ space^ $\tilde{M}$ of constant^ holomorphic^ curvature $\tilde{c}(\neq 0)$^ and^ let^ $H$^ be^ the
restricted holonomy^ group^ of $M$^ (with^ respect^ to^ the induced Kahlerian structure).
Then
i) if $c\sim<0,$^ $H$^ is always isomorphic to $U(n)$.
ii) (^) if $c\sim>0,$^ $H$^ is isomorphic either to $U(n)$^ or to $SO(n)\times S^{1}$^ , where $S^{1}$ denotes the circle group,^ the second case arising^ only^ when $M$^ is
locally holomorphically isometric to the complex quadric $Q^{n}$^ in $P^{n+1}(C)$^.
PROOF. i) Since $c\sim<0$^ , the Ricci tensor is negative definite according^ to
(8) and $M$^ is therefore nondegenerate^ (see^ [4];^ actually^ it was proved^ there
that $J\in H$ but the proof^ shows that $7\in \mathfrak{h}$^ ).^ Since $R(e_{i}, Je_{j})=(\frac{c\sim}{4}-\lambda_{i}\lambda_{j})S_{j}^{i}\in \mathfrak{h}$
and since $\lambda_{k}\geqq 0$^ for^ all^ $k$^ and^ $c\sim<0$^ ,^ we^ have^ $S_{j^{i}}\in \mathfrak{h}$^ for^ every^ pair^ $(i, j)$^.^ Since
$R(e_{i}, Je_{i})\in \mathfrak{h}$ and $J\in \mathfrak{h}$^ , we have $S_{i}^{i}\in \mathfrak{h}$. Thus $K_{j^{i}}=[S_{j}^{i}, S_{i}^{i}]\in \mathfrak{h}$^ for all $i,$^ $j$^.
Hence $\mathfrak{h}=\iota((n)$^.
ii) We first dispense with the case where $M$^ is an Einstein manifold,^ in
which case $A^{2}=\lambda^{2}I$. Since $\sum_{r=1}^{n}R(e_{\gamma}, Je_{r})=-\rho J\in \mathfrak{h}$^ ,^ where $\rho$^ is the Ricci curva-
ture of^ $M$,^ and^ since^ $\rho$^ is^ nonzero^ in^ view^ of^ Proposition^9 [8],^ we^ deduce
that $J\in \mathfrak{h}$^. From the curvature transformations $R(e_{i}, e_{j}),$^ $R(e_{i}, Je_{j})$^ and $R(e_{i}, Je_{i})$
we conclude^ that^ all^ $K_{j}^{i}(i\neq j)$^ and^ $S_{j^{i}}$^ ($i=j$^ included)^ are^ contained^ in^ $\mathfrak{h}$^ ,^ that
is, $H=U(n)$, unless $\lambda^{2}=\tilde{c}/4$^ (i. e. $\rho=nc\sim/2$). At any rate we know that $M$^ is
locally symmetric so that the curvature transformations at any^ point^ $x_{0}$
generate the holonomy^ algebra^ $\mathfrak{h}$^.^ If^ $\lambda^{2}=c\sim/4$^ ,^ we readily^ see that $\mathfrak{h}$ is generated
by $J$^ and by all $K_{j}^{i}$^ , that is $H=SO(n)\times S^{1}$^. On the other hand, the complex
quadric $Q^{n}=SO(n+2)/SO(n)\times SO(2)$ imbedded in $P^{n+1}(C)$^ with holomorphic
curvature $\tilde{c}$^ is^ Einstein^ and^ has^ holonomy^ group^ isomorphic^ to^ $SO(n)\times SO(2)$
(i. e. $SO(n)\times S^{1}$^ ). Thus $\lambda^{2}=c\sim/4$^ for $Q^{n}$^. Now if $\lambda^{2}=\tilde{c}/4$^ for $M$, the same
argument as was used in^ Proposition^11 of [8] can be applied locally to show
that $M$^ is locally^ holomorphically^ isometric^ to $Q^{n}$^. We have thus taken care
of Theorem 2 in^ the^ case^ where^ $M$^ is^ Einstein (getting^ a more precise result
than Proposition 10 of [8]).
If $M$^ is not^ an^ Einstein^ manifold^ we may^ assume that the characteristic
roots of $A^{2}$^ at $x_{0}$^ are^ not^ all^ equal.^ By^ (i)^ of Lemma 4 we know that $K_{\iota^{k}}\in \mathfrak{h}$
for all^ $k,$^ $l$^.^ If^ $ 4\lambda_{i}^{2}=c\sim$^ for^ some^ $i$^ ,^ then^ $R(e_{i}, Je_{i})=-\frac{c\sim}{2}J\in \mathfrak{h}$^. By the assumption
on $A^{2}$^ at^ $x_{0}$^ ,^ we have^ $4\lambda_{j}^{2}\neq\tilde{c}$^ for^ some^ $j$^ and consequently $S_{j}^{j}\in \mathfrak{h}$ from $R(e_{j}, Je_{j})$
$=-\frac{\tilde{c}}{2}J+2(\lambda_{j}^{2}-\frac{\tilde{c}}{4})S_{j}^{j}\in \mathfrak{h}$ . By (ii) of Lemma 4 we conclude that $\mathfrak{h}=u(n)$ , that
is, $H=U(n)$. We may therefore suppose $4\text{{\it \‘{A}}}{i}^{2}\neq\tilde{c}$ for every $i$. If $4\lambda{1}^{2}<\tilde{c}$^ , then
$ 4\lambda_{1}\lambda_{n}<c\sim$ , since $\lambda_{1}>\lambda_{n}$ ; therefore $R(e_{1}, Je_{n})\in \mathfrak{h}$ implies $S_{n^{1}}\in \mathfrak{h}$. By (iii) of Lemma
Differential geometry^ of complex^ hypersurfaces^ II^507
we say^ that Codazzi’s equation^ reduces. We have
LEMMA 5. The following^ conditions are equivalent^ on^ $M$^ :
i) Codazzi’s equation reduces.
ii) (^) The Ricci tensor (^) of $M$^ is parallel, (^) that is $\nabla S=0$^.
iii) $M$^ is locally symmetric.
REMARK. This result has been obtained independently^ by^ T. Takahashi
[9] using another method. In the case $c\sim\neq 0$^ we know by^ Theorem 2 in (^) \S 2
that either $M$^ is locally^ $Q^{n}$^ ,^ which is Einstein,^ or the holonomy^ group^ of $M$^ is
$U(n)$ . In the second case, $\nabla S=0$^ implies that $M$^ is Einstein because $M$^ is
irreducible. Thus Lemma 5 generalizes^ Theorem 2 of [8]^ only in the case
$\tilde{c}=0$ . We shall, however, give a direct proof of $(ii)\rightarrow(i)$ .
PROOF. The proof^ of^ Theorem 2 [8]^ shows that (i)^ implies^ (iii). (iii)
implies (ii) trivially. (^) We now show that (ii) implies (i). $\nabla S=0$^ is equivalent
to $\nabla A^{2}=0$^ and this in turn implies^ that the characteristic roots of $A^{2}$^ together
with their^ multiplicities^ are^ constant^ on^ $M$.^ Consequently,^ if^ $A^{2}=0$^ at one
point then $A^{2}$^ vanishes identically and Codazzi’s equation reduces trivially.
Assuming that this is not the case, let $\lambda_{1},$^ $\cdots$^ , $\lambda_{r}$ be the distinct positive
characteristic roots of $A$^ on $U(x_{0})$^. Consider the distributions on $U(x_{0})$^ defined
by $T_{i}^{+}(x)={X\in T_{x}(M)|AX=\lambda_{i}X}$ , $T_{i}^{-}(x)={X\in T_{x}(M)|AX=-\lambda_{\dot{t}}X}$ (^) , $T_{i}(x)=T_{i^{+}}(x)\oplus T_{i^{-}}(x)$ (^) ,
$T^{0}(x)={X\in T_{x}(M)|AX=0}$ .
Clearly $J$^ interchanges $T_{i^{+}}(x)$^ and $T_{i}^{-}(x)$^. (^) When $X$^ is an arbitrary vector field
and $Y$^ is a^ vector field^ in^ $T^{0}$^ we^ deduce^ from
$0=(\nabla_{X}A^{2})(Y)=\nabla_{X}(A^{2}Y)-A^{2}(\nabla_{X}Y)=-A^{2}(\nabla_{X}Y)$
that $\nabla_{X}Y\in T^{0}$^. Hence $T^{0}$^ is parallel.^ (A similar argument shows that each
$T_{i}$ is parallel.)
If $Y\in T^{0}$^ ,^ we have $(\nabla_{X}A)Y=\nabla_{X}(AY)-A\nabla_{X}Y=0$^. On the other hand, we
have $s(X)JAY=0$ so that^ $(\nabla_{X}A)Y=s(X)JAY$. By^ Codazzi’s equation (^) we also
obtain $(\nabla_{Y}A)X=s(Y)JAX$. In other words,^ the reduced Codazzi equation holds
when $X$^ or $Y$^ is in $T^{0}$^. Now $\nabla A^{2}=0$^ being^ equivalent to $(\nabla_{X}A)A+A(\nabla_{X}A)=0$
(for all $X$^ ), we see that $(\nabla_{X}A)T_{i^{+}}\subset T_{i}^{-}$^ and $(\nabla_{X}A)T_{i^{-}}\subset T_{i}^{+}$^. By (^) virtue of Codazzi’s
equation the reduced Codazzi equation holds for vector fields $X\in T_{i}$ and $Y\in T_{j}$
$(i\neq j)$ (^). We draw the same conclusion (^) when $X\in T_{i^{+}}$ (^) and $Y\in T_{i^{-}}$ (^) , or vice versa.
Finally, if $X,$^ $Y\in T_{i}^{+}$^ (or $T_{i^{-}}$^ ), then using $J(\nabla_{X}A)=-(\nabla_{X}A)J$^ and $JY\in T_{i^{-}}$^ we get
$(\nabla_{X}A)Y=-JJ(\nabla_{X}A)Y=J(\nabla_{X}A)JY=Js(X)JA(JY)=s(X)JAY$ .
In short, we have shown that the equation^ $(\nabla_{X}A)Y=s(X)JAY$^ holds for all
508 K.^ NOMIZU^ and^ B.^ SMYTH
$X,$ $Y$.
THEOREM 4. Let $M$^ be a complex hypersurface (^) of complex^ dimension^ $n\geqq 1$ in a space^ $\tilde{M}$ of constant^ holomorphic^ curvature^ $ c\sim$^.^ If the^ Ricci tensor^ of $M$^ is parallel, (^) then either $M$^ is (^) of constant holomorphic curvature $\tilde{c}$ and totally
geodesic in $\tilde{M}$ or $M$^ is locally^ holomorphically isometric to the complex^ quadric
$Q^{n}$ in $P^{n+1}(C)$ , the latter case arising only when $c\sim>0$ .
PROOF. When $n=1$^ the condition $\nabla S=0$^ simply means that $M$^ is of con-
stant curvature and the classification obtained in \S 6 will show that Theorem
4 is valid.
Assume $n\geqq 2$^.^ Let^ $c\sim\neq 0$^.^ In^ view of^ Lemma^ 5,^ $M$^ is locally^ symmetric.
Consequently, each $\tau\in H$, considered as parallel displacement of $T_{x0}(M)$^ along
a closed curve through^ $x_{0}$^ , maps^ the curvature tensor $R_{x_{0}}$^ at $x_{0}$^ into $R_{x0}$^. Thus
if $M$^ has restricted holonomy^ group^ $U(n)$^ then,^ since $U(n)$^ acts transitively^ on
the set of holomorphic^ planes^ at^ $x_{0}$^ ,^ we conclude^ that all^ holomorphic^ planes
at $x_{0}$^ have the same^ sectional^ curvature;^ since^ $x_{0}$^ is^ an^ arbitrary^ point,^ $M$^ has
constant holomorphic^ sectional^ curvature^ and^ immerses^ totally^ geodesically^ in
$\tilde{M}$ (see Theorem 1 [8]). If the restricted holonomy group of $M$ is not $U(n)$ ,
$M$ is locally holomorphically isometric to $Q^{n}$^ and $\tilde{c}>0$^ , by virtue of Theorem 2.
Let $\tilde{c}=0$^. The^ roots^ of^ $A^{2}$^ are^ constant^ in^ value and multiplicity^ on^ $M$,
since $\nabla A^{2}=0$^.^ Let^ us^ now^ suppose^ that^ $A^{2}\neq 0$^ and^ choose^ a^ basis^ { $e_{1},$^ $\cdots$^ ,^ $e_{n}$^ ,
$ Je_{1}\ldots$ , $Je_{n}$ } of $T_{x0}(M)$ diagonalizing $A$^ in the manner described in the previous
section. Using^ the computations^ of \S 2 and the fact that $\nabla R=0$^ and $c\sim=0$^ ,
we find $0=(R(e_{i}, e_{j})R)(e_{i}, Je_{j})=[R(e_{i}, e_{j}), R(e_{i},Je_{j})]-R(R(e_{i}, e_{j})e_{i},$ $Je_{j}$) $-R(e_{i}, R(e_{i}, e_{j})Je_{j})$ $=-\lambda_{i}^{2}\lambda_{j}^{2}[K_{j}^{i}, S_{j}^{i}]+\lambda_{i}\lambda_{j}R(e_{j}, Je_{j})-\lambda_{i}\lambda_{j}R(e_{i}, Je_{j})$ $=-2\lambda_{i}^{2}\lambda_{j}^{2}(S_{i}^{i}-S_{j}^{j})+2\lambda_{i}\lambda_{j}^{3}S_{j}^{j}-2\lambda_{i}^{3}\lambda_{j}S_{i}^{i}$ $=-2\lambda_{i}^{2}\lambda_{j}(\lambda_{i}+\lambda_{j})S_{i}^{i}+2\lambda_{i}\lambda_{j}^{2}(\lambda_{i}+\lambda_{j})S_{j}^{j}$ (^).
Thus $\lambda_{i}\lambda_{j}=0$^ or $\lambda_{i}+\lambda_{j}=0$^.^ Since $\lambda_{1}\geqq\lambda_{2}\geqq\ldots\geqq\lambda_{n}\geqq 0$^ and $\lambda_{1}>0,$^ $A^{2}$ has
precisely one nonzero characteristic^ root^ $\lambda_{1}^{2}$^ and^ its^ multiplicity^ is^ 2. We confine our attention to^ the^ distributions^ $T_{1}^{+},$^ $T_{1}^{-},$^ $T_{1}$^ and^ $T^{0}$^ on^ $U(x_{0})$^ ,^ as^ defined^ in Lemma 5.^ We have^ already^ seen^ that^ $T_{1}$^ and^ $T^{0}$^ are^ parallel^ on^ $M$^ and^ that
the reduced^ Codazzi^ equation^ holds^ by^ virtue^ of^ Lemma^ 5.^ Thus^ if^ $Z$^ is^ an
arbitrary vector and $W$^ is a^ unit vector field in $T_{1}^{+}$^ ,^ then
$s(Z)JAW=(\nabla_{Z}A)W=\nabla_{Z}(AW)-A\nabla_{Z}W=\lambda_{1}\nabla_{Z}W-A\nabla_{Z}W$ .
But since $T_{1}$^ is parallel^ and^ (real)^ 2-dimensional and $W$^ is a unit vector in
$T_{1}^{+}$ (^) , we see that $\nabla_{Z}W\in T_{1^{-}}$^ and $\text{\‘{A}}{1}\nabla{Z}W-A\nabla_{Z}W=2\lambda_{1}\nabla_{Z}W$. Therefore, (^) the
equation above reduces to^ $\lambda_{1}s(Z)JW=2\lambda_{1}\nabla_{Z}W$,^ that is,^ $\nabla_{Z}W=-2-s(Z)JW$. It
is an easy^ matter^ to^ verify^ that^ $R(X, Y)W=ds(X, Y)JW$,^ for^ arbitrary^ vectors
(^510) K. NOMIZU and B. SMYTH ii) (^) $D(Y, X)=-D(X, Y)$ , iii) $\mathfrak{S}{D(X, Y)Z}=0$ , where $\mathfrak{S}$ is the cyclic sum taken over $X,$^ $Y$^ and $Z$, iv) $\mathfrak{S}{(\nabla_{X}D)(Y, Z)}=0$^.
It is well known that the Riemannian curvature tensor field $R$^ of $M$^ satisfies
these conditions. We also note that (i),^ (ii)^ and (iii)^ imply
v) $g(D(X, Y)Z,$ $W$^ ) $=g(D(Z, W)X,$ $Y$^ ), as is the case for $R$^ (see [5], p. 198). We define^ the^ nullity^ space^ $T_{x}^{0}$^ of^ $D$^ at^ each^ point^ $x\in M$^ to^ be^ the^ subspace
{ $X|D(X,$^ $Y)=0$^ for^ all^ $Y\in T_{x}(M)$^ } of^ $T_{x}(M)$^ ;^ its^ dimension^ is^ called^ the^ index
of nullity^ of^ $D$^.^ Let^ $T_{x^{1}}$^ be^ the^ orthogonal^ complement^ of^ $T_{x^{0}}$^.^ The^ following lemmas can^ be^ proved^ in^ exactly^ the^ same^ way^ as those in^ [6].
LEMMA 6.
i) (^) If $X\in T_{x^{0}}$^ , then $D(Y, Z)X=0$ (^) for all $Y,$^ $Z\in T_{x}(M)$^.
ii) $T_{x^{1}}$ coincides with the subspace spanned by all $D(X, Y)Z$,^ where $X,$^ $Y,$^ $Z$
$\in T_{x}(M)$ .
LEMMA 7. Assume that the index (^) of nullity (^) of a curvature-type tensor (^) field
$D$ is constant on M. Then the distribution $T^{0}$^ : $x\rightarrow T_{x^{0}}$^ is involutive and totally
geodesic (that is, $\nabla_{X}T^{0}\subset T^{0}$ for any vector $X\in T^{0}$^ so (^) that any integral (^) manifold of $T^{0}$^ is^ a^ totally^ geodesic^ submanifold of $M$^ ).
We shall^ apply^ the^ foregoing^ lemma^ to^ the^ situation^ where^ $M$^ is^ a^ complex
hypersurface in a space^ $\tilde{M}$ of constant holomorphic^ curvature $ c\sim$. The curvature
tensor $R$^ of $M$^ is given^ by^ Gauss’ equation
$R(X, Y)=\tilde{R}(X, Y)+D(X, Y)$ (^) , the expressions for $\tilde{R}(X, Y)$^ and $D(X, Y)$ being as in (^) \S 1. Since both $R$^ and $\tilde{R}$ are curvature-type (^) tensor fields on $M$, (^) so is their difference $D$ (^). We (^) know (Lemma 1,^ \S 1) that the nullity^ space^ $T_{x^{0}}$ coincides with the kernel of $A$^ at $x$^.
Hence $\dim T_{x^{1}}$^ equals^ the^ rank of^ $M$^ at $x$^. Assume now that this is constant
on $M$. The distribution $T^{0}$^ is integrable and totally geodesic by Lemma 7; it
is also invariant by the complex structure $J$, because (^) $JA=-AJ$. If $M^{0}$^ is a
maximal integral^ manifold^ of^ $T^{0}$^ ,^ we conclude that $M^{0}$^ is a complex^ sub-
manifold of $M$^ which is totally geodesic in $M$. The curvature tensor $R^{0}$^ of $M^{0}$
(with respect to the metric induced from that of $M$^ ) is given by $R^{0}(X, Y)$
$=R(X, Y)$, where $X,$^ $Y\in T_{x}(M^{0})$^ , which is equal to $\tilde{R}(X, Y)$^ , since $D(X, Y)=0$
for $X,$^ $Y\in T_{x}(M^{0})=T_{x^{0}}$^.^ Thus $R^{0}(X, Y)=\frac{\tilde{c}}{4}{X\wedge Y+JX\wedge JY+2g(X, JY)J}$ (^) , which means that^ $M^{0}$^ has^ constant^ holomorphic^ curvature $\tilde{c}$.
Differential geometry^ of complex^ hypersurfaces^ II^511
Considering $M^{0}$^ as a complex submanifold of $\tilde{M}$^ ,^ we may^ establish the formula $\tilde{K}(X)=K^{0}(X)+2\sum_{i=1}^{k}{g(A_{i}X, X)^{2}+g(JA_{i}X, X)^{2}}$
(for a unit vector $X$^ tangent to $M^{0}$^ ) relating the sectional curvatures $\tilde{K}(X)$
and $K^{0}(X)$^ in $\tilde{M}$ and $M^{0}$^ , respectively, of the holomorphic plane generated by
X. In this formula $A_{1},$^ $\cdots$^ , $A_{k}$^ are the second fundamental forms corresponding
to a choice of an orthonormal family of vector fields $\xi_{1},$^ $\cdots$^ , $\xi_{k}$ normal to $M^{0}$^ ,
and $k$ is the complex codimension of $M^{0}$^ in $\tilde{M}$. This formula generalizes that
of Corollary 2 [8]. Since $\tilde{K}(X)=K^{0}(X)=c\sim$^ in our case,^ it follows that each
$A_{i}$ is identically zero, which means that $M^{0}$^ is totally geodesic in $\tilde{M}$.
Let us (^) now assume that $M$^ is a complete complex hypersurface in $P^{n+1}(C)$^ ,
$C^{n+1}$ or $D^{n+1}$ such that the rank of $M$ is everywhere equal to $2r$ . We show
that $M^{0}$^ is then complete. Let $\gamma(s)$^ be a geodesic in $M^{0}$^ defined on $a<s<b$.
Since $M$^ is complete^ and^ $M^{0}$^ is^ totally^ geodesic^ in^ $M,$^ $\gamma(s)$^ can be extended as
a geodesic $\gamma^{*}(s)$^ in $M$, defined for all values of $s$^. Let $(x^{1}, \cdots , x^{2m}, x^{2m+1}, \cdots , x^{2n})$^ ,
where $m=n-r$,^ be^ a^ system^ of^ local^ coordinates on $M$^ with origin^ $\gamma^{*}(b)$^ ,^ such
that ${\frac{\partial}{\partial x^{1}}$^ ,^ $\cdot$^ ..^ ,^ $\frac{\partial}{\partial x^{2m}}}$ is a^ local basis for $T^{0}$^. When $s$^ is in a certain neigh-
borhood of $b$^ , say $(b-\epsilon, b+\epsilon)$^ , we may express $\gamma^{*}(s)$^ by the set of equations
$x^{i}(\gamma^{}(s))=f^{i}(s),$ $1\leqq i\leqq 2n$ (^). Since $\gamma^{}(s)=\gamma(s)cM^{0}$ (^) when $a<s<b$ we must then have $f^{i}(s)=c^{i}$^ (a constant) for $2m+1\leqq i\leqq 2n$^. Letting $s$^ approach $b$ from
below we find that $0=f^{i}(b)=c^{i},$^ $2m+1\leqq i\leqq 2n$^. Thus $\gamma^{*}(b)$^ is^ in^ the^ maximal
integral manifold which contains $\gamma(s),$^ $a<s<b$. In other words $\gamma^{*}(b)\in M^{0}$
and it is possible to extend $\gamma(s)$^ as a geodesic in $M^{0}$^ for parameter values
larger than $b$. Thus $M^{0}$^ is complete.
Since we know that any^ complete^ totally^ geodesic^ complex^ n-dimensional
submanifold of $P^{n+1}(C),$^ $C^{n+1}$^ ,^ or^ $D^{n+1}$^ is^ of^ the^ form^ $P^{m}(C),$^ $C^{m},$^ $D^{m}$,^ respectively,
we obtain
PROPOSITION 1. Let $M$^ be^ a^ complex^ hypersurface^ of $\tilde{M}=P^{n+1}(C),$^ $C^{n+1}$^ ,^ or
$D^{n+1}$ . If the rank (of the second fundamental form) of $M$ is everywhere equal
to a^ constant,^ $2r$^ ,^ then^ $M$^ contains^ a^ complete^ totally^ geodesic^ complex^ $(n-r)-$
dimensional (^) submanifold of $\tilde{M}$^ ,^ namely^ $P^{n- r}(C),$^ $C^{n-r},$^ $D^{n- r}$^ ,^ respectively. We now prove^ the main theorem of^ this^ section
THEOREM 6. Let $M$^ be a^ compact^ complex^ hypersurface^ of $P^{n+1}(C),$^ $n\geqq 3$^.
The rank (of^ the second (^) fundamental form)^ of $M$^ cannot^ be^ identically^ equal
to 2.
REMARK. For $n=1$^ ,^ the^ quadrics^ are^ the^ only^ closed^ complex^ curves^ in
$P^{2}(C)$ of rank identically equal to 2 (see (i) of Theorem 9 in \S 6). The case
$n=2$ remains unsettled.
Differential geometry^ of complex^ hypersurfaces^ II^513
\S 5. Hypersurfaces^ in^ $C^{n+1}$^.
To begin^ with,^ we suppose^ that $M$^ is a complex^ hypersurface^ in^ an
arbitrary K"ahlerian^ manifold $\tilde{M}$. For^ any^ vector field $X$^ on $M$^ and^ for^ any
field of vectors $\xi$ normal to $M$^ in $\tilde{M}$^ ,^ we define $\hat{\nabla}_{X}\xi$ to be the normal component
of $\tilde{\nabla}_{X}\xi$ , where $\tilde{\nabla}$^ refers,^ as in [8],^ to covariant differentiation in $\tilde{M}$. We may
easily verify that $\hat{\nabla}$ is a linear connection in the normal bundle over $M$, which
we call^ the^ normal^ connection^ for^ the^ hypersurface^ $M$.^ The^ relative^ curvature
tensor $\hat{R}$ of $M$^ (that^ is,^ the curvature tensor of^ the normal connection of $M$^ )
is given^ by
$\hat{R}(X, Y)\xi=[\hat{\nabla}{X},\hat{\nabla}{Y}]\xi-\hat{\nabla}_{[X,Y]}\xi$ (^) ,
where $X$^ and^ $Y$^ are^ vector^ fields^ tangent^ to^ $M$.^ If^ $\xi$^ is^ a^ field^ of^ unit^ normals,
$\hat{\nabla}_{X}\xi$ is equal to $ s(X)J\xi$ and, by an easy computation, we find
PROPOSITION 2. The relative curvature tensor $\hat{R}$ of $M$^ is expressed by $\hat{R}(X, Y)\xi=2ds(X, Y)J\xi$ (^) , where $\xi$ is a (^) field of unit normals to $M$.
Now assume that $\tilde{M}$ has constant holomorphic^ sectional curvature $ c\sim$.
According to Proposition^4 of [8],^ we have
$\tilde{S}(X, JY)=S(X, JY)+2ds(X, Y)$ (^) , where $\tilde{S}$ and $S$^ denote the Ricci tensors of $\tilde{M}$ and $M$, respectively. We shall
prove
THEOREM 7. Let $M$^ be a complex hypersurface (^) of complex (^) dimension $n\geqq 1$ in a space^ $\tilde{M}$ of constant holomorphic curvature $ c\sim$. The following conditions are equivalent: i) The normal connection (^) of $M$^ is trivial,^ that is, $\hat{R}=0$^.
ii) $S=\tilde{S}$^ on $M$.
iii) $S=0$^ on $M$.
iv) $c\sim=0$^ and $M$^ is totally geodesic in $\tilde{M}$.
PROOF. It is clear that iv) implies each of the other conditions, while the
equivalence of i) and ii) follows from Proposition 2 above. Assuming ii) we
see that $M$^ is Einstein. By Theorem 4,^ $M$^ is then totally geodesic in $\tilde{M}$ or
else $c\sim>0$^ and $M$^ is locally holomorphically isometric to $Q^{n}$^ in $P^{n+1}(C)$ . Thus
$S=(n+1)-\frac{c\sim}{2}g$ or else $c\sim>0$ and $S=\frac{n\tilde{c}}{2}g$ . However, $\tilde{S}=(n+2)\frac{c\sim}{2}g$. Therefore
$\tilde{c}=0$ and $S=0$ and consequently $M$ is totally geodesic in $\tilde{M}$. In other words,
ii) implies both iii) and iv). If $S=0$, then $M$^ is Einstein and it is clear from
the above that $c\sim=0$^ and $M$^ is totally geodesic in $\tilde{M}$. Thus iii) implies iv) and
the equivalence of all four conditions is proved.
The general object of the remainder of this section is to define the Gaussian
514 K. NOMIZU and B. SMYTH
mapping of a complex hypersurface in complex Euclidean space $C^{n+1}$^ into the
complex projective space $P^{n}(C)$^ , and to give a geometric interpretation thereof.
It is convenient to begin by establishing a relationship between the Riemannian
connection on the sphere $S^{2n+1}$^ and the K"ahlerian connection on $P^{n}(C)$ (for the
Fubini-Study metric, of course).
$P^{n}(C)$ can be regarded as the base of a principal fibre bundle $S^{2n+1}$^ (unit
sphere (^) in $C^{n+1}$^ ) (^) on which the structure group $S^{1}={e^{i\theta}|\theta\in R}$^ acts as follows: $S^{2n+1}\times S^{1}\ni(z, e^{t\theta})\rightarrow ze^{i\theta}\in S^{2n+1}$ (^). $\pi$ (^) denotes the canonical projection of $S^{2n+1}$
onto $P^{n}(C)$^ and $g(z, w)={\rm Re}(\sum_{k=0}^{n}z^{k}w^{-k})$^ for $z=(z^{0}, z^{1}, \cdots , z^{n}),$^ $w=(w^{0}, w^{1}, \cdots , w^{n})$
defines the Euclidean metric on $C^{n+1}$^. With the natural identification between
vectors tangent^ to $S^{2n+1}$^ and vectors $C^{n+1}$^ , we have
$T_{z}(S^{2n+1})={w\in C^{n+1}|g(z, w)=0}$
for each $z\in S^{2n+1}$^. The orthogonal complement of
$T_{z^{\prime}}={w\in C^{n+1}|g(z, w)=g(iz, w)=0}$
in $T_{z}(S^{2n+1})$^ is the l-dimensional subspace^ ${iz}$^ which is spanned by the vector
$iz$ (in the sense of the above identification). The distribution $T^{\prime}$ defines a
connection in the principal^ fibre bundle^ $S^{2n+1}(P^{n}(C), S^{1})$^ ,^ that is,^ $T_{z}^{\prime}$ is comple-
mentary to the subspace^ ${iz}$^ tangent^ to^ the^ fibre^ through^ $z$^ ,^ and $T^{\prime}$ is invariant
by the action of $S^{1}$. Thus the projection $\pi$ induces a linear isomorphism of
$T_{z}^{\prime}$ onto $T_{\pi(z)}(P^{n}(C))$ and $\pi$ maps ${iz}$ into zero for each $z\in S^{2n+1}$ .
The classical Fubini-Study metric of holomorphic sectional curvature 1
is nothing^ but the metric $\tilde{g}$ defined by^ $\tilde{g}(\tilde{X},\tilde{Y})=4g(X^{\prime}, Y^{\prime})$ , where $\tilde{X},\tilde{Y}$
$\in T_{p}(P^{n}(C))$ (^) and $X^{\prime},$^ $Y^{\prime}$ are their respective (^) horizontal lifts at $z(\pi(z)=p)$ (^).
Since $g$^ is invariant by^ $S$^ ‘,^ the^ definition^ of^ $\tilde{g}(\tilde{X},\tilde{Y})$^ is^ independent^ of the
choice of $z$^. We might also observe that the complex structure in $T_{z}^{\prime}$ (defined
by multiplication of vectors by i) induces the canonical complex structure $J$^ on
$P^{n}(C)$ , when transferred by means of $\pi$^. (What we have said so far is more
or less well known.)
The horizontal lift of a vector field $\tilde{X}$ on $P^{n}(C)$^ will be denoted by $X^{\prime}$.
If $\tilde{X}$ and $\tilde{Y}$ are vector fields^ on^ $P^{n}(C)$^ ,^ then^ the^ vector fields $X^{\prime}$ and $Y^{\prime}$ are
invariant by^ $S^{1}$^ ;^ since the^ Riemannian^ connection^ on^ $S^{2n+1}$^ is invariant by^ $S^{1}$^ ,
it follows that $\nabla_{X^{\prime}}^{\prime}Y^{\prime}$^ (where^ $\nabla^{\prime}$^ denotes^ covariant^ differentiation^ on^ $S^{2n+1}$^ )^ is
also invariant by^ $S^{1}$^ and^ hence^ projectable,^ that^ is,^ there^ exists^ a^ vector field
$\tilde{Z}$ on $P^{n}(C)$ such that $\pi_{*}(\nabla_{X^{\prime}}^{\prime}Y^{\prime}){z}=\tilde{Z}{\pi(z)}$ for all $z\in S^{2n+1}$ .
PROPOSITION 3. For every pair^ of vector (^) fields $\tilde{X},\tilde{Y}$ on $P^{n}(C)$^ the vector field $\nabla_{X}^{\prime},Y^{\prime}$^ on^ $S^{2n+1}$^ is^ projectable^ and^ $\tilde{\nabla}{\tilde{X}}\tilde{Y}=\pi{*}(\nabla_{X^{\prime}}^{\prime}Y^{\prime})$ defines the^ Kahlerian
connection on $P^{n}(C)$^.
PROOF. To^ prove^ this^ we^ verify^ the^ following:
i) $\tilde{\nabla}$ is a linear connection. Obviously $\tilde{\nabla}_{\tilde{X}}\tilde{Y}$ is bi-additive in $X$^ and $Y$.
(^516) K. (^) NOMIZU and B. (^) SMYTH
Thus $\phi(x)=\pi(\xi)\in P^{n}(C)$^ is well defined and the mapping $\phi$ : $M\rightarrow P^{n}(C)$ is
called the Gaussian mapping of $M$. We can relate $\phi$ to the second fundamental
form $A$^ of $M$^ (in the formalism of [8]) as follows:
Let $X\in T_{x}(M)$ (^) and take a curve $x_{t}$ (^) on $M$^ such that $x_{0}=x$ and $(\vec{x}{c}){t=0}=X$.
Choose a (differentiable) family of unit normals $\xi_{t}$ along $\chi_{/}$ . The differential
$\Phi*of\phi$ maps $X$ upon
$(\frac{d\pi(\xi_{\iota})}{dt}){t=0}=\pi{*}(\frac{d\xi_{t}}{dt}){t=0}\in T{\phi(x)}(P^{n}(C))$ (^) ,
where $(\frac{d\xi_{t}}{dt}){t=0}$^ is the tangent^ vector of the curve $\xi{t}$ on $S^{2n+1}$^ at $\xi_{0}$. On the
other hand,^ the Weingarten formula for $M$^ as a complex hypersurface in $C^{n+1}$
(with the flat connection $D$^ ) gives
$(\frac{d\xi_{t}}{dt}){\iota=0}=D{X}\xi=-AX+s(X)J\xi$ , where $ J\xi=i\xi$ .
Since $ J\xi$^ is the initial tangent^ vector of the curve $ e^{i\theta}\xi$ on $S^{2n\vdash 1}$^ ,^ we have
$\pi_{*}(J\xi)=0$ . Hence
$\phi_{}(X)=-\pi_{}(AX)$ .
The (^) vector $AX$, considered by translation as a tangent (^) vector to $S^{2n+1}$^ at $\xi$ ,
belongs to $T_{\xi^{\prime}}$ because it is perpendicular to $ J\xi$^. Since $\pi_{*};$ $T_{\xi^{\prime}}\rightarrow T_{\pi(\xi)}(P^{n}(C))$^ is
one-to-one, we conclude that
i) $\phi_{}(X)=0$^ if and only (^) if $AX=0$. ii) (^) The rank (^) of $\Phiis$^ equal to the rank (^) of $A$^.
Since $\phi_{}(JX)=-\pi_{}(AJX)=\pi_{*}(JAX)$^ and^ since^ the^ complex^ structure^ $J$^ on
$T_{\xi^{\prime}}$ corresponds to the complex structure $J$^ on $T_{\pi(\xi)}(P^{n}(C))$ , by means of $\pi$ , we
have
$\phi_{}(JX)=\tilde{J}\pi_{}(AX)=-\tilde{J}\phi_{*}(X)$ (^) , namely,
iii) the Gaussian mapping $\phi$ is anti-holomorphic.
EXAMPLES.
i) If $M$^ is a hyperplane $C^{n}$^ in $C^{n+1}$^ we have a constant unit normal $\xi$ over
$M$, so that $\phi(M)$ is a single point in $P^{n}(C)$ .
ii) If $M$^ is of the form $K\times C^{n-1}$^ , where $K$^ is a complex curve in a plane
$C^{2}$ perpendicular to $C^{n-1}$ , then the rank of $\phi$ is $\leqq 2$ everywhere and $\phi(M)$ lies
in a projective^ line $P^{1}(C)$^ in $P^{n}(C)$^. It will be interesting^ to find an appropriate
converse of^ this^ proposition.
In relating the K"ahlerian^ connection on $M$^ to that on $P^{n}(C)$^ , the following
lemma will be useful.
LEMMA 9. Let $x_{t}$^ be a differentiable curve on M. Then there is a family
of unit^ normals^ $\xi_{t}$^ along^ $x_{t}$^ which,^ as^ a^ curve^ in^ $S^{2n+1}$^ ,^ is^ horizontal.
PROOF. For an arbitrary family of unit normals $\eta_{t}$^ along $x_{t}$^ we consider
Differential geometry^ of complex^ hypersurfaces^ II^517
a family of unit normals given^ by^ $\xi_{t}=a\eta_{t}+bJ\eta_{t}$^ ,^ where $a=a(t)$^ and^ $b=b(t)$
are differentiable functions such that $a^{2}+b^{2}=1$^. We show that by^ choosing^ $a$
and $b$^ suitably we can make $\xi_{t}$^ horizontal,^ that is $g(\frac{d\xi_{t}}{dt},$^ $J\xi_{t})=0$^ for all $t$.
It is^ readily^ verified^ that
$g(\frac{d\xi_{t}}{dt},$ ]$\xi_{t})=g(\frac{d\eta_{t}}{dt},$ $J\eta_{t})+a\frac{db}{dt}-b\frac{da}{dt}$ .
Thus our^ purpose^ will^ be^ achieved if^ we^ can^ choose^ $a$^ and^ $b$^ such^ that
$a\frac{db}{dt}-b\frac{da}{dt}=k(t)$ (^) and $a^{2}+b^{2}=1$ (^) ,
where $k(t)=-g(\frac{d\eta_{t}}{dt}$^ ,^ $J\eta_{t})$^.^ Since^ $a^{2}+b^{2}=1$^ implies^ $a\frac{da}{dt}+b\frac{db}{dt}=0$^ ,^ we^ have
$\frac{da}{dt}=-bk(t)$ and $\frac{db}{dt}=ak(t)$ .
Thus we may take
$a(t)=\cos l(t)-\sin l(t)$ , $b(t)=\sin l(t)+\cos l(t)$ (^) ,
where $1(t)=\int k(f)dt$^.
THEOREM 8. Let $M$^ be a complex^ hypersurface^ in^ $C^{n+1}$^ and let^ $Y_{\iota}$ be a
family (^) of vectors tangent (^) to $M$^ along a (^) curve $x_{t}$^. Choose a family (^) of unit
normals $\xi_{t}$ along $x_{t}$^ as in Lemma 9 and let $Y_{t^{\prime}}$ be the vector tangent to $S^{2n+1}$^ at
$\xi_{t}$ which is parallel to $Y_{t}$ in $C^{n+1}$ . Let $\tilde{Y}{t}=\pi{*}(Y_{t}^{\prime})$. Then $Y_{t}$ is parallel along
$x_{t}$ on $M$^ if and only if $\tilde{Y}{t}$ is parallel along $\phi(x{t})$^ on $P^{n}(C)$^.
PROOF. For $M$^ we have
(10) $\frac{dY_{t}}{dt}=D_{x_{t}}^{A}Y_{t}=\nabla_{x_{t}}^{\rightarrow}Y_{t}+h(\rightarrow x_{t}, Y_{t})\xi_{t}+k(\rightarrow x_{t}, Y_{t})J\xi_{t}$^ ,
where $D$^ is the flat^ connection in^ $C^{n+1}$^ and^ $\nabla$^ is^ the^ K"ahlerian^ connection^ on
$M$. On the other hand, for $S^{2n+1}$^ (with the Riemannian connection $\nabla^{f}$ ) we get
(11) $\frac{dY_{t}}{dt}=\frac{dY_{t}^{1}}{dt}=D_{\xi_{t}^{\rightarrow}}Y_{t}^{\prime}=\nabla_{\rightarrow,\xi_{t}}^{\prime}Y_{t}^{\prime}+h^{\prime}(\xi_{t}\rightarrow, Y_{t}^{\prime})\xi_{t}$ ,
where $h^{\prime}$^ is^ the^ second^ fundamental^ form^ of^ $S^{2n+1}$^ with^ respect^ to^ the^ unit
normals $\xi_{t}$. Equations (10) and (11) yield
$\nabla_{x_{t}}^{\rightarrow}Y_{\ell}+{h(\rightarrow x, Y_{t})-h^{\prime}(\xi_{t}\rightarrow, Y_{t}^{\prime})}\xi_{\iota}+k(\overline{x}{t}, Y{t})J\xi_{t}=\nabla_{\xi_{i}}^{\prime}\rightarrow Y_{t^{\prime}}$
\langle considered as an identity among^ vectors in $C^{n+1}$^ ). Therefore
$\nabla_{x_{t}\iota_{\xi_{t}}}^{\rightarrow Y=\nabla^{\prime}\rightarrow Y_{t}^{\prime}-k(x_{t}}\rightarrow,$ $Y_{t}$)$J\xi_{t}$^.
Thus, if $\nabla_{x_{t}}^{\rightarrow}Y_{t}=0$^ , the fact that both $\rightarrow\xi_{t}$ and $Y_{t}^{\prime}$ are horizontal and that $J\xi_{r}$^ is
vertical in^ $T_{\xi\iota}(S^{2n+1})$^ implies
