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Let $(M,\langle \cdot,\cdot\rangle)$ be a pseudo-Riemannian manifold and $f: M \to \Bbb R$ be a smooth function. One can consider the covariant Hessian $\nabla ({\rm d}f)$. Some time ago I had seen a …

I'm reading the paper Classification of marginally trapped Lorentzian flat surfaces in $\mathbb{E}^4_2$ and its applications to biharmonic surfaces by B. Y. Chen. Summarizing it quickly: he first pr …

If $V$ is a vector space and $g$ is a symmetric degenerate bilinear form on $V$, every complementary subspace to the radical ${\rm rad}(V)$ is called a "screen subspace" of $V$: we have an orthogonal …

Let $1 \leq \nu \leq n+1$ and $M^n \subseteq \Bbb S^{n+2}_\nu$ be a non-degenerate submanifold. Assume that $\renewcommand{\vec}[1]{{\bf #1}} \vec{L}_0 \in \Bbb R^{n+3}_\nu$ is lightlike and $M \subse …

I am trying to read the paper Marginally Trapped Submanifolds in Space Forms with Arbitrary Signature by Henri Anciaux, but I think that there is a mistake in Lemma $1$, in page $5$: The second fu …

Question: I'd like to know if there is some reference or reasonable way to develop curve theory in a plane with degenerate metric $(\Bbb R^2, {\rm d}s^2 ={\rm d}x^2)$. Context: In Lorentz-Minkowski s …

Let $\alpha \colon I \to \Bbb R^2_1$ be a regular curve and $t_0 \in I$ be such that $\alpha$ is lightlike at $t_0$, and not lightlike at $]t_0-r,t_0[$ for some $r>0$. Then, in that interval the curva …

We know that for $n\geq 2$ the de Sitter space $\mathbb{S}^n_1(r)$ and the hyperbolic space $\mathbb{H}^n(r)$ have constant curvature $1/r^2$ and $-1/r^2$, respectively. Looking at references such as …