Ngô Quốc Anh

January 31, 2010

R-G: Gauss and Codazzi equations in Riemannian geometry

Filed under: Riemannian geometry — Ngô Quốc Anh @ 10:29

So far, we have discussed Codazzi equations in classical differential geometry. Today, we discuss the Gauss and Codazzi equations in Riemannian geometry. Later on, we will discuss in a new entry the Gauss and Codazzi equations in general relativity.

A classical problem in geometry is to determine whether a Riemannian manifold $M$ can be isometrically immersed in another Riemaniann manifold $\overline M$. We will restrict ourselves to the case of codimension $1$ immersions, i.e., $M$ has dimension $n$ and $\overline M$ has dimension $n + 1$.

It is well known that the Gauss and Codazzi equations are necessary conditions relating the Riemann curvature tensor $R$ of $M$, the Riemann curvature tensor $\overline R$ of $\overline M$ and the shape operator $S$ of $M$ (or the second fundamental form). Denoting by $\nabla$ the Riemannian connection of $M$, these equations are the following $\displaystyle\left\langle {R\left( {X,Y} \right)Z,W} \right\rangle- \left\langle {\overline R \left( {X,Y} \right)Z,W} \right\rangle= \left\langle {SX,Z} \right\rangle \left\langle {SY,W} \right\rangle- \left\langle {SY,Z} \right\rangle \left\langle {SX,W} \right\rangle$

and $\displaystyle {\nabla _X}SY - {\nabla _Y}SX - S\left[ {X,Y} \right] = \overline R \left( {X,Y} \right)N$

for all vector fields $X$, $Y$, $Z$ and $W$ on $M$.