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Proof of Laugwitz Conjecture and Landsberg Unicorn Conjecture for Minkowski norms with
$SO(k)\times SO(n-k)$-symmetry
Published online by Cambridge University Press: 03 June 2021
Abstract
For a smooth strongly convex Minkowski norm
$F:\mathbb {R}^n \to \mathbb {R}_{\geq 0}$
, we study isometries of the Hessian metric corresponding to the function
$E=\tfrac 12F^2$
. Under the additional assumption that F is invariant with respect to the standard action of
$SO(k)\times SO(n-k)$
, we prove a conjecture of Laugwitz stated in 1965. Furthermore, we describe all isometries between such Hessian metrics, and prove Landsberg Unicorn Conjecture for Finsler manifolds of dimension
$n\ge 3$
such that at every point the corresponding Minkowski norm has a linear
$SO(k)\times SO(n-k)$
-symmetry.
Keywords
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- Copyright
- © Canadian Mathematical Society 2021
Footnotes
The first author is supported by Beijing Natural Science Foundation (No. Z180004), NSFC (No. 11771331 and No. 11821101), and Capacity Building for Sci-Tech Innovation—Fundamental Scientific Research Funds (No. KM201910028021). The second author thanks DFG for partial support via projects MA 2565/4 and MA 2565/6.
References
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