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another fix
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mateuszbaran committed Dec 4, 2024
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4 changes: 3 additions & 1 deletion tutorials/what-are-manifolds.qmd
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Expand Up @@ -160,7 +160,9 @@ It has the following properties:

* The function $F$ is smooth on $T \mathcal{M} \setminus \{0\}$.
* For all $p\in \mathcal{M}$, $X \in T_p \mathcal{M}$ and $\lambda \geq 0$ the metric is homogeneous: $F(p, \lambda X) = \lambda F(p, X)$.
* Strong convexity: at each $p\in \mathcal{M}$ the Hessian of $X \mapsto \frac{1}{2}F^2(p, X)$ is positive definite[^no-strong-convexity]. This Hessian $g_{p}$ is called the fundamental tensor.
* Strong convexity: at each $p\in \mathcal{M}$ the Hessian of $X \mapsto \frac{1}{2}F^2(p, X)$ is positive definite[^no-strong-convexity].
This Hessian $g_{p}$ is called the fundamental tensor.

For each point $p$ the function $X \mapsto F(p, X)$ is a Minkowski norm on $T_p\mathcal{M}$, that is the following properties hold:

* Positivity: $F(p, X) > 0$ for all $X \neq 0$,
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