Integral curves in null hypersurfaces [closed]

Let be $$(M^{n+1},g)$$ a spacetime (Lorentz manifold, connexe and time-oriented), $$n\ge 2$$, and $$S\subset M$$ a null hypersurface (codim $$S=1$$ and the restriction of $$g$$ to each tangent space $$T_p S$$ is degenerate).

If $$K$$ is a null vector field of $$S$$, show that the integral curves of $$K$$ are null geodesics of $$S$$.

Everyone has a hint?

ERRATA: I would like understand why that problem is equivalent to show that $$\nabla_K K=\lambda K,$$

where $$\lambda\in C^\infty(S)$$ is a smooth function. Cause, by $$\nabla_K K=\lambda K$$, if $$\alpha$$ is a integral curve of $$K$$, $$\frac{d\alpha}{dt}=K(\alpha(t))$$, then

$$\frac{D}{dt}\Big(\frac{d\alpha}{dt}\Big)=K(\alpha (t)),$$

and I don't understand why the right-hand side is zero.

Maybe I found something like that in the book General Relativity: With Applications to Astrophysics, By Norbert Straumann, p. 273.

It's sufficient to consider $$\nabla \psi = K$$, I think. But I don't understand the final line, in short the notation.

Thanks again.

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