Let us first look at the real case. Suppose is a real Riemannian manifold of real dimension with Riemannian metric . Let be a submanifold of with real dimension . We choose local coordinates of so that is given by the vanishing of . We give the metric induced from that of . So its Riemannian metric tensor is also with . Let us take a local orthonormal frame of tangent vectors of so that at points of , the vectors are tangential to . Let be the dual frame of so that is a -form on . Let be the Levi-Cevita connection of expressed as a matrix-valued -form in terms of the frame . We want to determine the Levi-Civita connection of . For a vector field on , we define covariant derivative in as the orthogonal projection of the covariant derivative onto the tangent space of . Clearly this connection is compatible with the metric, because if vanishes along a curve of , then along is perpendicular to , and the derivative of the square of the length of along is equal to twice the inner product of with along and must vanish. The matrix valued -form representing the connection of is simply . To verify that the connection of is torsion-free, we have to check that the skew-symmetrization of the covariant derivative of agrees with the exterior derivative of . In other words,
vanishes on , where means covariant differenation with respect to the connection on . We know that
vanishes on , where means covariant differentiation with respect to the connection . The vanishing of on follows from the fact that vanishes identically on . Thus, we know that the connection on is the Levi-Civita connection.
We now compare the curvature tensor on and the curvature tensor on . We have
The difference of the two curvature tensors is the term . We are going to interpret this term in terms of what is called the second fundamental form of .
Let denote the tangent bundle of . It is a subbundle of the tangent bundle of . Let be the normal bundle of in which is defined as the quotient bundle over . We identify with the orthogonal complement of in . Suppose we have two vector fields on . The difference is an element of the normal bundle of in . For any smooth functions on , we have
This means that at a point depends only on the values of and at . Let us denote at a point depends only on the values of and at . Let us denote by . Then is a section of over and is called the second fundamental form of . In terms of the local frame , we have
The second fundamental form is symmetric in and . To check this at a point , we can choose local coordinates so that is defined by the vanishing of and
at point . Then equals the Christoffel symbol , which is symmetric in and because of the torsion-free condition of the Levi-Civita connection.
We can regard as an -valued -form on . Let denote the -valued -form on which is the adjoint of with respect to the metrics of and . Then
can be rewritten as
as -valued -forms. Even though 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 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 vanishes at a point, the curvature of agrees with that of . Suppose is constructed in the following way from a linear subspace of the tangent space of at a point . For every , we construct the geodesic through in the direction of . The totality of as varies in forms a submanifold in a neighborhood of , and this manifold is . The second fundamental form of is identically zero at , because is clearly zero at and is linear in and . In the special case when is spanned by two tangent vectors
Our manifold is a surface, and we denote it by . Then the Gaussian curvature of is
is the square of the norm of the -vector in the metric induced from the Riemannian metric . One verifies this by first considering the case when and 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
of is known as the Riemannian sectional curvature of the plane spanned by and .