If all eigenvalues of a hypersurface in Euclidean space with dimension \geq 3, are equal at a point, then such hypersurface is totally umbilical at that point: $A_p=\lambda(p) Identity$.

When all the points of a hypersurface are umbilical, by the Codazzi equation, the function $\lambda$ is (locally) constant, and then, it is an open part of either a totally geodesic hyperplane or a sphere of radius $1/\labmda$.

I agree with the answer given above.

Distinguished professors, thank you very much for your answer.

I agree with the answers as well. I have an additional question. For a surface immersed in Euclidean space, such that its principal curvatures differ by a constant, is there a name you know of? Of course, if $k_1=k_2$, the surface would be totally umbilical. What if $k_1-k_2= constant$? Thanks! :-) This relates to an older question of mine and I was just wondering about terminology and meanings. Equivalently what is the name of a surface with property $H^2 - K = constant$? Other than some special quadratic Weingarten surface....

For the complete classification of totally umbilical submanifolds in indefinite real space forms, you can find in my book "Pseudo-Riemannian geometry, [delta]-invariants and applications, pp. 57-62" (World Scientific, 2011).

I have read Professor Chen's reference which has answers to above question and more information on this issue

 Dear Professor Chen,

After asking this question, I had the chance to study your helpful book while researching. We have this book in our university libraries.

Thank you very much for your interest.

Dear Profesor Duggal,

Thanks for concerning my question too.

Sincerely.

yes I liked it very much

Respected Prof. Chen and Prof. Duggal

I wish to say that on going through your conversations, I get motivated to study the Book of Prof. Chen. We have not the copy of this book in our library but will try to arrange soon.

Regards.

Dear Prof. Chen,

I liked it too and I still study your book. (And I will study)

With warmest regards,

Alev Kelleci

Wish you very Happy holidays and all the best in new year 2016 and best of luck in your study