For a Riemannian manifold $(M,g)$
Given a geodesic segment $\gamma:[a,b]\to M$ we define a symmetric bilinear form $I$ , called the index form of , on the space of normal vector fields along $\gamma$ by:
I think The sign of I gives me the following infomation about $\gamma$.It tells me that : if I(V,V)<0 for every proper normal vector field along $\gamma \implies \gamma $ is not minimizing. (this from corollary 10.23). Lee's book says that the index form should be like a hessian,so I would have expected that if it is negative, then the geodesic is maximazing (and not sure if a maximizing geodesic makes sense)
Is that it? Moreover I find this confusing because aren't geodesics by definition minimizing curves? Lee's book says that the index form should be like a hessian, so I was expecting it would tell me something also when the index is positive, like a positive index implies the geodesic is minimizing But the (contrapositive of the) corollary only tells me what I wrote above, since it is not an iff
