1-introduction this article has a title of " Multi-class Brain Coputer interface By Riemannian Geometry" its authors are:Alexendre Barachant, Stéphane Bonnet (CEA-LETI Campus MINATEC Grenoble France )and Marco Congedo, Christian Jutten (GIPSA Lab, CNRS, Université de Grenoble, France) it is extracted in the journal, IEEE Transactions on Biomedical Engineering, Institute of Electrical and Electronics Engineers (IEEE), 2012, 59 (4), pp.920-928. . computer science [cs], signal and image processing, engeneer science [physics] PubMed ID: 22010143 INSPEC Accession Number : 12614948 Submitted on 21 Mar 2012 sources: https://hal.archives-ouvertes.fr/hal-00681328wikipedia Proc. IEEE Toronto Int. Conf. Sci. Technol. Humanité (TIC-STH) , pp. 268-273, 2009. Alexendre Barachant is a Research Scientist, specialized in Machine Learning for Brain-Computer Interfaces & Neuroscience, in Burke-Cornell Medical Research Institute in New York specialised in EEG signals and BCI, Christian Jutten is PHD (Doctor-ingeneer) at Institut National Polytechnique de Grenoble (INPG) addressed study of artificial neural networks as information processing systems. Marco Congedo is a scientist research at National Research Institute in Computer Science and Control (INRIA) in France. Stéphane Bonnet is research team leader at Becton Dickinson Medical Pharmaceutical systems. According to Google Scholar this article has been cited 126 times, the authors presented us two methods of classification framework for BCI(Brain Computer Interface), based on Motor Imagery(MI); that they will be compared to a referencial method.So in the first, authors started by prensing us the method of BCI classification (spatial Filters applications) exposing the most fields BCI are used for.After That, we will see how to use the covarience matrices in BCI.Then we have some definitions to understend more the context of the article. Next, the two methods are explained and their results are analysed , the Minimum Distance to Riemannian Mean (MDRM) and the Tangent Space Linear Discriminent Analysis (TSLDA). Finaly, their contributions and contraints are deduced and the estimations and predictions of the use of these methods in diferent fields.2-Some definitions: BCI: brain computer interface is a direct communication interface which links humain brain with hardware (computer or electronic components) it used to assist, improve or repair human function of failing congnition or action. we hope to use it in medical application for exemple to restaure lost faculties (such as hearing, sight, or even mouvements) and why not to achieve to improve those faculties more then their naturel level (so we speak about exocortex)"EEG:is a cerebral exploration method that measures the electrical activity of the brain by electrodes placed on the scalp often represented in the form of a trace called electroencephalogram" from wikipedia .Motor Imagery: is the mentale representation of a mouvement or physical action in order to improve or make the excecution of the mouvement stable.3-Context and Positioning:BCI aims to translate brain signals to commandes; it can be used for handicap persons in order to restore communication and mouvements(Medical application), in the other part; use it in video game applications for the normal personnes(Non-Medical application).In this article the BCI's control is based on MI but it can be used for other types of BCI. EEG experiences are long and complicated, that’s why it evolved to a simplification of the procedure.