A characterization of hyperbolic spaces
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초록
Let $M$ be a complete spacelike hypersurface in the $(n+1)$-dimensional Minkowski space ${\mathbb L}^{n+1}$. Suppose that every unit speed curve $X(s)$ on $M$ satisfies $\<X''(s), X''(s)\>\ge -1/r^2$ and there exists a point $p\in M$ such that for every unit speed geodesic $X(s)$ of $M$ through the point $p$, $\<X''(s), X''(s)\>= -1/r^2$ holds. Then, we show that up to isometries of ${\mathbb L}^{n+1}$, $M$ is the hyperbolic space $H^n(r)$.
Abstract
Let $M$ be a complete spacelike hypersurface in the $(n+1)$-dimensional Minkowski space ${\mathbb L}^{n+1}$. Suppose that every unit speed curve $X(s)$ on $M$ satisfies $\<X''(s), X''(s)\>\ge -1/r^2$ and there exists a point $p\in M$ such that for every unit speed geodesic $X(s)$ of $M$ through the point $p$, $\<X''(s), X''(s)\>= -1/r^2$ holds. Then, we show that up to isometries of ${\mathbb L}^{n+1}$, $M$ is the hyperbolic space $H^n(r)$.
참고문헌 (3)
- J. Baek, "A characterization of the unit sphere", Amer. Math. Monthly, 110(9), pp.830~833 (2003)[DOI]
- D.-S. Kim, "On standard imbeddings of hyperbolic spaces in the Minkowski space", C. R. Math. Acad. Sci. Paris, 352(12), pp.1033~1038 (2014)[DOI]
- B. O'Neill, "Semi-Riemannian Geometry, Pure and Applied Mathematics, 103" (1983), Academic Press, Inc.