# Connections, The Kähler Condition, and Curvature – VII. The Second Fundamental Form in the Riemannian Case

Let us first look at the real case. Suppose ${M}$ is a real Riemannian manifold of real dimension ${m}$ with Riemannian metric ${g_{ij}}$. Let ${S}$ be a submanifold of ${M}$ with real dimension ${s}$. We choose local coordinates ${x^1, \dots, x^m}$ of ${M}$ so that ${S}$ is given by the vanishing of ${x^{s+1}, \dots, x^m}$. We give ${S}$ the metric induced from that of ${M}$. So its Riemannian metric tensor is also ${g_{ij}}$ with ${1 \le i,\, j \le s}$. Let us take a local orthonormal frame ${e_i}$ ${(1 \le i \le m)}$ of tangent vectors of ${M}$ so that at points of ${S}$, the vectors ${e_i}$ ${(1 \le i \le s)}$ are tangential to ${S}$. Let ${f^i}$ ${(1 \le i \le m)}$ be the dual frame of ${e_i}$ ${(1 \le i \le m)}$ so that ${f^i}$ is a ${1}$-form on ${M}$. Let ${\omega_i^j}$ ${(1 \le i,\,j \le m)}$ be the Levi-Cevita connection of ${M}$ expressed as a matrix-valued ${1}$-form in terms of the frame ${e_i}$ ${(1 \le i \le m)}$. We want to determine the Levi-Civita connection of ${S}$. For a vector field ${X}$ on ${S}$, we define covariant derivative ${\nabla^S X}$ in ${S}$ as the orthogonal projection of the covariant derivative ${\nabla^M X}$ onto the tangent space of ${S}$. Clearly this connection is compatible with the metric, because if ${\nabla^S X}$ vanishes along a curve ${\gamma}$ of ${S}$, then ${\nabla^M X}$ along ${\gamma}$ is perpendicular to ${S}$, and the derivative of the square of the length of ${X}$ along ${\gamma}$ is equal to twice the inner product of ${X}$ with ${\nabla^M X}$ along ${\gamma}$ and must vanish. The matrix valued ${1}$-form representing the connection of ${S}$ is simply ${(\omega_i^j)_{1 \le i,\, j \le s}}$. To verify that the connection of ${S}$ is torsion-free, we have to check that the skew-symmetrization of the covariant derivative of ${f^i|S}$ agrees with the exterior derivative of ${f^i|S}$. In other words,

$\displaystyle D_\wedge^S f^i - df^i = \sum_{j=1}^s \omega_j^i \wedge f^j$

vanishes on ${S}$, where ${D^S}$ means covariant differenation with respect to the connection on ${S}$. We know that

$\displaystyle D_\wedge^M f^i - df^i = \sum_{j=1}^m \omega_j^i \wedge f^j$

vanishes on ${M}$, where ${D^M}$ means covariant differentiation with respect to the connection ${M}$. The vanishing of ${\sum_{j=s+1}^m \omega_j^i \wedge f^j}$ on ${S}$ follows from the fact that ${f^j}$ ${(s + 1 \le j \le m)}$ vanishes identically on ${S}$. Thus, we know that the connection ${(\omega_i^j)_{1 \le i, j \le s}}$ on ${S}$ is the Levi-Civita connection.

We now compare the curvature tensor ${\Omega^S}$ on ${S}$ and the curvature tensor ${\Omega^M}$ on ${M}$. We have

$\displaystyle \Omega_i^{M\text{ }j} = d\omega_i^j - \sum_{k=1}^m \omega_i^k \wedge \omega_k^j$

and

$\displaystyle \Omega_i^{S\text{ }j} = d\omega_i^j - \sum_{k=1}^s \omega_i^k \wedge \omega_k^j.$

Thus,

$\displaystyle \Omega_i^{S\text{ }j} = \Omega_i^{M\text{ }j} + \sum_{k=s+1}^m \omega_i^k \wedge \omega_k^j.$

The difference of the two curvature tensors is the term ${\sum_{k=s+1}^m \omega_i^k \wedge \omega_k^j}$. We are going to interpret this term in terms of what is called the second fundamental form of ${S}$.

Let ${T_S}$ denote the tangent bundle of ${S}$. It is a subbundle of the tangent bundle ${T_M}$ of ${M}$. Let ${N_{S, M}}$ be the normal bundle of ${S}$ in ${M}$ which is defined as the quotient bundle ${T_M/T_S}$ over ${S}$. We identify ${N_{S,M}}$ with the orthogonal complement of ${T_S}$ in ${T_M|S}$. Suppose we have two vector fields ${X, Y}$ on ${S}$. The difference ${\nabla_Y^M X - \nabla_Y^M X}$ is an element of the normal bundle ${N_{S, M}}$ of ${S}$ in ${M}$. For any smooth functions ${\varphi, \psi}$ on ${S}$, we have

$\displaystyle \nabla_{\psi Y}^M(\varphi X) - \nabla_{\psi Y}^S(\varphi X) = \varphi \psi(\nabla_Y^M X - \nabla_Y^S X).$

This means that ${\nabla_Y^M X - \nabla_Y^S X}$ at a point ${P}$ depends only on the values of ${X}$ and ${Y}$ at ${P}$. Let us denote ${\nabla_Y^M - \nabla_Y^S X}$ at a point ${P}$ depends only on the values of ${X}$ and ${Y}$ at ${P}$. Let us denote ${\nabla_Y^M X - \nabla_Y^S X}$ by ${\Pi(X, Y)}$. Then ${\Pi}$ is a section of ${N_{S, M} \otimes T_S^* \otimes T_S^*}$ over ${S}$ and is called the second fundamental form of ${S}$. In terms of the local frame ${e_i}$ ${(1 \le i \le m)}$, we have

$\displaystyle \Pi(e_i, e_j) = \sum_{k=s+1}^m \omega_i^k(e_j)e_k.$

The second fundamental form ${\Pi(X, Y)}$ is symmetric in ${X}$ and ${Y}$. To check this at a point ${P}$, we can choose local coordinates ${x^1, \dots, x^m}$ so that ${S}$ is defined by the vanishing of ${x^{s+1}, \dots, x^m}$ and

$\displaystyle {\partial\over{\partial x^i}} = e_i \text{ }(1 \le i \le m)$

at point ${P}$. Then ${\omega_i^k(e_j)}$ equals the Christoffel symbol ${\Gamma_{ij}^k}$, which is symmetric in ${i}$ and ${j}$ because of the torsion-free condition of the Levi-Civita connection.

We can regard ${\Pi}$ as an ${\text{Hom}(T_S, N_{S, M})}$-valued ${1}$-form on ${S}$. Let ${\Pi^*}$ denote the ${\text{Hom}(N_{S, M}, T_S)}$-valued ${1}$-form on ${S}$ which is the adjoint of ${\Pi}$ with respect to the metrics of ${T_S}$ and ${N_{S, M}}$. Then

$\displaystyle \Omega_{i}^{S\text{ }j} = \Omega_i^{M\text{ }j} + \sum_{k = s+1}^m \omega_i^k \wedge \omega_k^j$

can be rewritten as

$\displaystyle \Omega^S = \Omega^M + \Pi \wedge \Pi^*$

as ${\text{End}(T_S)}$-valued ${2}$-forms. Even though ${\Pi \wedge \Pi^*}$ is the product of a matrix and its adjoint, we can not conclude that it has a fixed sign, because the product is an exterior product. So the curvature of a submanifold can be greater than or less than that of the ambient manifold. However, as we will see later, it is possible to get a fixed sign for ${\Pi \wedge \Pi^*}$ is some cases, and certain curvatures of a complex submanifold are always no greater than those of the ambient complex submanifold.

When the second fundamental form of ${S}$ vanishes at a point, the curvature of ${S}$ agrees with that of ${M}$. Suppose ${S}$ is constructed in the following way from a linear subspace ${V}$ of the tangent space of ${M}$ at a point ${P}$. For every ${v \in V}$, we construct the geodesic ${\gamma_v}$ through ${P}$ in the direction of ${v}$. The totality of ${\gamma_v}$ as ${v}$ varies in ${V}$ forms a submanifold in a neighborhood of ${P}$, and this manifold is ${S}$. The second fundamental form ${\Pi}$ of ${S}$ is identically zero at ${P}$, because ${\Pi(v, v)}$ is clearly zero at ${P}$ and ${\Pi(X, Y)}$ is linear in ${X}$ and ${Y}$. In the special case when ${V}$ is spanned by two tangent vectors

$\displaystyle \xi = \xi^i {\partial\over{\partial x^i}},\text{ }\eta = \eta^i {\partial\over{\partial x^i}}.$

Our manifold ${S}$ is a surface, and we denote it by ${S_{\xi,\eta}}$. Then the Gaussian curvature of ${S_{\xi,\eta}}$ is

$\displaystyle {{R(\xi, \eta, \xi, \eta)}\over{|\xi \wedge \eta|^2}},$

where

$\displaystyle R(\xi, \eta, \xi, \eta) = R_{ijk\ell}\xi^i \eta^j \xi^k \eta^\ell$

and

$\displaystyle |\xi \wedge \eta|^2 = {1\over2}g^{ik}g^{j\ell}(\xi^i \eta^j - \eta^i \xi^j)(\xi^k \eta^\ell - \eta^k \xi^\ell)$

is the square of the norm of the ${2}$-vector ${\xi \wedge \eta}$ in the metric induced from the Riemannian metric ${g_{ij}}$. One verifies this by first considering the case when ${\xi}$ and ${\eta}$ are orthonormal and using the above relation between the Gaussian curvature of a surface and the curvature tensor defined as a measure of failure of the commutativity of partial covariant differentiation. The Gaussian curvature

$\displaystyle {{R(\xi, \eta, \xi, \eta)}\over{|\xi \wedge \eta|^2}}$

of ${S_{\xi, \eta}}$ is known as the Riemannian sectional curvature of the plane spanned by ${\xi}$ and ${\eta}$.