Flatness of a statistical manifold with Fisher information metric Let $\mathcal{M} = \{p_\theta := p(\cdot | \theta), \theta \in \Theta\}$ be a statistical manifold with Fisher information metric: $$g_{{jk}}(\theta )=\operatorname {E} \left[\left({\frac {\partial }{\partial \theta _{i}}}\log p(X;\theta )\right)\left({\frac {\partial }{\partial \theta _{j}}}\log p(X;\theta )\right) \right].$$
The Wikipedia article on the topic derives the metric form Euclidean metric by changing variables. I can understand the procedure but I have questions related to the flatness of $\mathcal{M}$. In Amari's book; "Information Geometry and its application", it is said that such manifold is flat  (dually flat actually) so
1- Is the above derivation enough to conclude that the manifold is flat (I mean the fact that the metric is derived from the Euclidean metric)?
2- Is there a straightforward way to show that the curvature is 0 everywhere?
 A: This is a common source of confusion.
The flatness in information geometry refers to the dual (with respect to the Fisher-Rao metric) connections $\nabla$ and $\nabla^*$, called exponential and mixture connections, and not to the metric or Levi-Civita connection. For the latter, please see
Bruveris, M., & Michor, P. W. (2019). Geometry of the Fisher-Rao metric on the space of smooth densities on a compact manifold. Mathematische Nachrichten, 292(3), 511–523. https://doi.org/10.1002/mana.201600523
Finite-dimensional statistical manifolds can have negative curvature, for example the information manifold of 1-dimensional Gaussians is isometric to the Poincaré half-plane.
A: You just have to read Jean-Louis Koszul Work who has developed elementary structure of Information Geometry. You will find main Koszul references in the paper "Jean-Louis Koszul and the elementary structures of Information Geometry":
https://www.academia.edu/attachments/56029291/download_file?st=MTUyNTYzMDQ1NiwxOTIuNTQuMTQ0LjIyOSw0OTY2MjA3Mw%3D%3D&s=swp-toolbar
Fisher metric could be extended on homogeneous manifolds or on Lie group by Fisher-Souriau metric developed in the framework of Souriau Lie Group Thermodynamics:
[A] Marle, C.-M. From Tools in Symplectic and Poisson Geometry to J.-M. Souriau’s Theories of Statistical Mechanics and Thermodynamics. Entropy 2016, 18, 370.
http://www.mdpi.com/1099-4300/18/10/370/pdf
[B] Barbaresco, F. Geometric Theory of Heat from Souriau Lie Groups Thermodynamics and Koszul Hessian Geometry: Applications in Information Geometry for Exponential Families. Entropy 2016, 18, 386.
http://www.mdpi.com/1099-4300/18/11/386/pdf
Other references:
GSI’17 Geometric Science of Information
Third International Conference, GSI 2017, Paris, France, November 7-9, 2017, Proceedings
Editors: Nielsen, Frank, Barbaresco, Frédéric (Eds.)
http://www.springer.com/us/book/9783319684444
Videos: https://www.youtube.com/channel/UCnE9-LbfFRqtaes49cN2DVg/videos
CIRM seminar TGSI'17 on Topological & Geometrical Structures of Information
http://forum.cs-dc.org/category/94/tgsi2017
Differential Geometrical Theory of Statistics
Frédéric Barbaresco and Frank Nielsen (Eds.)
Pages: XIV, 458, Published: 6 June 2017
(This book is a printed edition of the Special Issue Differential Geometrical Theory of Statistics that was published in Entropy)
Free Download: http://www.mdpi.com/books/pdfdownload/book/313/1
Information, Entropy and Their Geometric Structures
Frédéric Barbaresco and Ali Mohammad-Djafari (Eds.)
Pages: XXIV, 528, Published: 1 September 2015
(This book is a printed edition of the Special Issue Information, Entropy and Their Geometric Structures that was published in Entropy)
Free Download: http://www.mdpi.com/books/pdfdownload/book/127/1 
