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}.

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