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.

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