# Angle between two vectors in a Minkowski (Finsler) space!

Given a Minkowski (or Finsler) space $(V,F)$, I am wondering how to define the angle between two vectors $w$ and $v$. I first thought it must be as $$\cos\theta(w,v)=\frac{g_w(w,v)}{\sqrt{g_w(w,w)g_w(v,v)}}.$$ Indeed I fixed the inner product $g_w$ and then defined the angle using it. I am also suspected that one can define the angle as $$\cos\theta(w,v)=\frac{g_w(w,v)}{F(v)F(w)}.$$

Any idea?

P.S. by $g_w$ I mean the second fundamental form which is defined as $$g_w(v,u)=\frac{1}{2}\frac{\partial^2 F^2}{\partial t \partial s}(w+su+tv)|_{s=t=0}.$$

• I doubt that you need angle in Finsler space. If you insist, then look at Riemannian case, choose the most important property of angle and use it as the definition --- typically different properties lead to different definitions. The one you propose looks okey, it is not symmetric, but you always have to sacrifice something in the Finsler world. – Anton Petrunin Dec 20 '17 at 18:57
• @AntonPetrunin I need the angle in the case of Finsler since I am trying to show that the angle between two special vectors remain constant along a submanifold. Could you please give me some reference where the angle in Riamannian is defined. You mentioned that the one that I defined looks OK. Which one? I have defined two. – Majid Dec 20 '17 at 19:38
• @AntonPetrunin You mentioned that different properties lead to different definitions. Is there any property which leads to my second definition? – Majid Dec 20 '17 at 19:39
• I agree with @AntonPetrunin. Following Busemann, I would also say: do you need vectors? Perhaps define the angle at $p$ between curves $a$ and $b$ as $2 \lim \arcsin d(a_s, b_s)/2s$, where $a_s$ and $b_s$ are the points at distance $s$ from $p$ along the curves. – Matt F. Dec 20 '17 at 19:43
• @Majid, is not it the first variation formula if you move in the direction $v$ and measure the distances to a point in the direction $w$? – Anton Petrunin Dec 20 '17 at 20:15