Dear Arkady:

Lie sphere geometry describes the geometry of oriented hyper-spheres in  euclidean n-sphere S^n, including single points, viewed as degenerated spheres. Here are the steps to Lie geometry.  You consider S^n \subset R^{n+1} \subset RP^{n+1}. A (hyper-)sphere can be viewed as a point p in the outer space E^{n+1} = RP^{n+1}\setminus D^{n+1}. In fact,  the tangent cone C_p with vertex p, consisting of tangent lines of  S^n passing through p, determines a hypersphere C_p \cap S^n, see attached figure. We may put  p = [x]  for some  x \in R^{n+2}  with = 1 with respect to the Lorentzian inner product

      = x_1y_1+...+x_{n+1}y_{n+1} - x_{n+2}y_{n+2};

note that S^n = \{[x]: \ = 0\}  and  E^{n+1} = \{[x] : > 0\}. There is precisely one pair \pm x  of such points with p = [\pm x],  for any p \in E^{n+1}. These two vectors x,-x describe a little more that just p, namely the two possible orientations on the sphere determined by p (inner or outer normal vector). Thus the Lorentzian symmetric space  L = \{x \in R^{n+1,1} : = 1\}  describes the space of oriented spheres in S^n (Moebius geometry). Lie sphere geometry adds a boundary bd(L) \cong S^n to this space. This is obtained again by embedding L into projective space RP^{n+2}. Then the inhomogeneous equation  - 1 = 0  describing  L has to be homogenized to  -x_{n+3}^2 = 0. The expression on the left hand side defines a new inner product now on R^{n+3}, say 

    ' = z_1^2 + ... + z_{n+1}^2 - z_{n+2}^2 - z_{n+3}^2

which has index 2, and the closure \bar L of L is

     \bar L = \{[z] : z\in R^{n+3}, z \neq 0,  ' = 0\}.

This object (an indefinite symmetric space) describes the Lie spheres: oriented spheres including those of radius 0 which are given by  bd(L) = \bar L \setminus L (the ideal points of \bar L). The symmetry group of Lie sphere geometry is thus G = PSO(n+1,2), acting on RP^{n+2} and keeping  \bar L  invariant. The spheres of radius 0 play no exceptional role any more; they are transformed by G also into spheres with positive radii.

This is what I know about Lie sphere geometry. I hope it helps answering your question.

Best regards

Jost

Dear Jost, many-many  thanks for your very thoughtful and detailed answer. While I was waiting for somebody's reply, I've run across this book 

http://www.amazon.com/Geometry-Hypersurfaces-Springer-Monographs-Mathematics/dp/1493932454/ref=sr_1_1?ie=UTF8&qid=1459277470&sr=8-1&keywords=geometry+of+hypersurfaces    It was just published in 2016. It does contain a lot of what is needed. But your answer is also fantastic. I would be glad to tell you more about why  I need this information.

Dear Prof Arkady, Here is a nice book on Lie sphere geometry:

"Thomas E. Cecil: Lie Sphere Geometry - With Applications to Submanifolds, Springer 1992.

Originally, Lie defined a map from line geometry to sphere geometry.  Somehow, the declining interest in line geometry turned that into just the study of sphere geometry.  Since I have an interest in line geometry, I have made English translations of the early papers on the subject by Lie, Klein, and Study.  If you are interested, my translations are posted at neo-classical-physics,info as free PDF downloads.

Dear Mukut,         Yes, I am familiar with the book by Cecil but I found that book Geometry of Hypersurfaces more helpful.It is much more user friendly. Nevertheless,I would be very happy to look as well  at originals translated by David. I've looked at neoclassical  physics web site.It is overwhelming. David, you deserve the highest award for such unique effort! I am absolutely speechless. In out times to have person like yourself,  is the same thing as to find signs of life on, say, Pluto. My infinite respect to you!!!

Interestingly (for me), when Lie developed his sphere geometry he also developed a concept of Lagrangian manifolds.This concept is used in contact geometry/topology problematics, e.g. look at my book http://www.worldscientific.com/worldscibooks/10.1142/8514

However, neither Cecil  in the book mentioned by Mukut nor Cecil and Ryan in their newest book make any link with applications to contact geometry/topology.  This application was on mind of Sofus Lie in the first place while no books on contact geometry and topology mention Lie's spherical geometry.Not even works by Vladimir Arnol'd. Strange...Correct me, please, if I am wrong.

Once again Many-many thanks to all of you: Jost, Mukut and David !  

Thank you for those inspiring words of encouragement, Arkady.

Lie's book with Scheffers "Beruhrungstransformationen" (contact transformations) has a 70 page chapter in which he discusses the line-sphere transformation.  I have had other queries about getting at least a translation of that chapter, so I might move it up my priorities list.

Pure mathematicians seem to have a tendency to burn their bridges behind them; i.e., once they get the new way of saying things, they stop talking about the old way.  Very few younger mathematicians would even realize that contact transformations are based in the concept of "things in space" contacting each other to various orders (intersection, tangency, osculation...), which also relates to the more modern concept of jets, and contact transformations preserve that relationship.

 Dear David, your contribution to science is impossible to estimate. It is priceless.!!! Yes, mathematicians tend to burn their bridges.But, remember, Bourbaki volumes. They all have historical remarks/background keeping records straight about  who did what and when. Vladimir Arnol'd did not like Bourbaki series of books because he considered these volumes as too formal. Well, I am just the opposite.I like Bourbaki style, especially for their efforts to keep record straight. Many of today's mathematicians reinvent the wheel in such a way that it is impossible to comprehend what exactly prompted them to do this or that. Books are devoid of any hints /seeds of the original idea. Too bad!  Hopefully, your efforts will be appreciated by many younger people.I am not young any more, unfortunately. But I feel young again when I am reading your wonderful translations. 

Dear Professor,

Please check the second edition of my book, Lie Sphere Geometry, Springer, 2008, ISBN 978-0-387-74655-5.

Yours sincerely,

Tom Cecil ([email protected])

College of the Holy Cross

Many thanks Thomas. So, as an author, you realize, that the 1st edition of your book is not as good as you would like it to be.  I am surely going to look into 2nd edition. In the meantime, I was able to find a book by Bobenko and Suris.Appendix to this book contains excellent introduction to the subject http://www.amazon.com/Discrete-Differential-Geometry-Graduate-Mathematics/dp/0821847007/ref=sr_1_1?ie=UTF8&qid=1459433976&sr=8-1&keywords=bobenko Also, I was able to find a book on advanced geometry by Felix Klein. This book surely makes everything crisply clear. So, if your 2nd edition is not as good as Klein's treatment of Lie's spherical geometry, I would recommend you to write the 3rd edition following style of Klein. For me, it would be of interest to elaborate on connections between contact geometry/topology and spherical geometry.Klein makes an attempt to do so  while books by Arnol'd, and the most up to date  http://www.amazon.com/Introduction-Topology-Cambridge-Advanced-Mathematics/dp/0521865859/ref=sr_1_3?ie=UTF8&qid=1459434286&sr=8-3&keywords=contact+geometry+and+topology have absolutely no mention of these Lie results.Shame! Once again,many-many thanks.

Dear Arkady,

Thank you for your comments.

Best regards,

Tom

U can write google lie sphere geometry

Cenap, of course, I used google too. However, math sci net is actually better for this purpose

Dear prof.  Arkady Kholodenko, 

you 'll be very grateful if you send me an electronic version of the monograph :

Geometry of Hypersurfaces (Springer Monographs in Mathematics) 1st ed. 2015 Edition by Thomas E Cecil (Author), Patrick J. Ryan (Author)

For its part, can offer a review of the scope of the three-dimensional hypersurfaces .

https://www.researchgate.net/publication/271486882_GS_Melnikov_Fraktalnoe_edinstvo_prostranstva_i_vremeni_Gennady_S_Melnikov_Space-time_fractal_unity

Three-dimensional sphere are built with the assistance of the unit quaternion parametric constructions involving Clifford algebra. In advance I express my appreciation to you .

Gennady Semenovich.

Book Г.С. Мельников Фрактальное единство пространства и времени. ...

Gennady, I cannot transfer to you copy righted material.But, I know that in in Russia, you guys are using many pirat sites to get what you need. Just try to use one of them.If not, we shall think about other options

Спасрсибо за ответ. Хаккерскими способностями не

Nikakih hakerskih sposobnostei ne nuzhno vovse.Hotite po drugomu, poshlite mne e-mail [email protected]

Dear Arkady think Could you send me E-mail:

[email protected]

and your monograph

http://www.worldscientific.com/worldscibooks/10.1142/8514

 ,

for its part to give you the opportunity to mathematically decrypt drawings, stated on its cover :

https://www.researchgate.net/publication/271671961_Sposob_sozdania_vremennyh_zaderzek_svetovogo_potoka_-_patent_RF_2109257

(file:///C:/Users/User/Downloads/patent2[1].pdf)

Уважаемый Аркадий   думаю, что Вас не затруднит выслать мне на E-mail

[email protected]

и вашу монографию

  http://www.worldscientific.com/worldscibooks/10.1142/8514

Я же со своей стороны даю Вам возможность математически расшифровать  рисунки, приведённые на  её обложке:

https://www.researchgate.net/publication/271671961_Sposob_sozdania_vremennyh_zaderzek_svetovogo_potoka_-_patent_RF_2109257

(file:///C:/Users/User/Downloads/patent2[1].pdf)

Patent Способ создания временных задержек светового потока - патент...

İ have a book. İts title is subject of your question. İ am studying dual numbers and vectors. These subjects are close to each other. Mainly, we have to know clifford algebra. Besides lie group structure, plucker coordinates etc. are very important.

Ferhat, what is the title of your book? I found also a number of sources other than book by Cecil. Book by Cecil is good when you know the subject already. The book I found is this http://www.amazon.com/Conformal-Differential-Geometry-Generalizations-Mathematics/dp/0471149586/ref=sr_1_1?s=digital-text&ie=UTF8&qid=1459949658&sr=8-1&keywords=conformal+geometry+goldberg

Surely, I can read this book from a to z . No problems.After reading this book it is possible to penetrate through Cecil's book.