What are the different geometries that are defined on space of matrices, especially the general linear group
You can use this link :
https://books.google.com/books?id=SO1oNBjt5hMC&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false
There is a very well-developed theory of the geometry of semi-simple Lie groups that includes the geometry of the special linear group. One passes to the corresponding symmetric space by quotienting out by the maximal compact subgroup (SO(n) for the group SLn and studies the geometry of the resulting space equipped with a natural metric coming from the Killing form. Here is the standard reference:
https://books.google.co.in/books?hl=en&lr=&id=j2Y71tZCTZUC&oi=fnd&pg=PP8&dq=eberlein+Geometry+of+Non+Positively+Curved+Manifolds&ots=P8IDClY3QK&sig=eKRh7ZWyga1njM5jcvYGPOUN5w8#v=onepage&q=eberlein%20Geometry%20of%20Non%20Positively%20Curved%20Manifolds&f=false
Dear Muhammad
Any affine space can be considered as the semidirect sum of the general linear Lie algebra and a vector space as an abelian Lie algebra.
Greetings, Tadeusz
There is a distinguished inner product on the space R^{nxn} of real nxn matrices: = (1/n) trace (A^T B) where A^T denotes the transposed matrix. The transposition and all left and right multiplications with orthogonal matrices (in O_n) are isometric with respect to this inner product (which is the usual euclidean inner product on R^{n^2}, up to the constant 1/n). Moreover, the orthogonal group O_n lies in the unit sphere S \subset R^{nxn} for this inner product. This has many interesting consequences. E.g. some great spheres in S actually lie in O_n; these are essentially the representations of the Clifford algebra (spinors).
There are analogues for complex and quaternion matrices.
Best regards
Jost
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