Beijing Institute of Mathematical Sciences and Applications

Shalabh Gautam , Xiaoming John Zhang

Fotsing Tetsing Hans

Thursday, June 25, 2026 2:00 PM - 3:00 PM

A3-4-101

Zoom 204 323 0165 (BIMSA)

Let $x:\Sigma\to M$ be a lightlike submanifold (also called null submanifold or degenerate submanifold) of a semi-Riemannian manifold $(M, g)$. The radical bundle $Rad(T\Sigma):=T\Sigma\cap T\Sigma^\perp$, where $T\Sigma$ and $T\Sigma^\perp$ denote respectively the tangent bundle and the normal bundle to $\Sigma$, is nontrivial, i.e. different from $\{0\}$. We say that $\Sigma$ is a $1-$lightlike submanifold if $\dim Rad(T_x\Sigma)=1$. In this case, $Rad(T\Sigma)$ is spanned by a null vector field, say $\xi$, which is everywhere tangent and orthogonal to $\Sigma$. It is clear that all null hypersurfaces are $1-$lightlike submanifolds. In particular, black hole horizons are examples of $1-$lightlike submanifolds.

• We will discuss riggings on a $1-$lightlike submanifold, which share many of the appealing properties of the Gauss map for nondegenerate hypersurfaces.
• We will show that the only null Monge hypersurfaces (i.e. graph of a function) $x:\Sigma\to\mathbb R^{n+2}_1$ satisfying the eigenvalue equation $\Delta x=\lambda x$ are null hyperplanes.
• Using null hypersurfaces, we will prove that a smooth function $f:\mathbb R^{n+1}\supset \mathcal O\to\mathbb R$ is affine if and only if $f$ is harmonic and has a
  gradient of constant norm.
• We will see how to use \href{https://www.sagemath.org/download.html}{SageMath}
  to work on manifolds.