Hi, I am reading the lecture notes on Morse Homology written by M.Hutchings, in that notes definition of Hessian is defined in coordinate free way: given any connection $ H(f,p)= \nabla_v(df)$ where $v$ is the tangent vector at critical point $p$, and $df$ is differential of $f$. I need to show this definition does not depend on choice of connection. Hutchings says that the difference of two connection is a tensor and $df$ vanishes at p, so the above fact holds. I can not understand the meaning of "the difference of two connection is a tensor" and how this observation solves my problem.

Thanks,