Dear Professor Yadav, I can not really interprete your question. I think that the space of all embeddings of S^{n-1} to S^n is path-connected (maybe up to some trivialities).

Yours B.Shapiro

For instance, check

A. Ros, Compact hypersurfaces with constant higher order mean curvatures.

Rev. Mat. Iberoamericana 3 (1987), no. 3-4, 447–453

  @ Prof  Boris Shapiro,    

Thanks a lot Prof Boris Shapiro. I want to know what are the structures on this class and their properties.

@ Prof. Miguel Ortega,

Thanks a lot Prof. Miguel Ortega. I need to study thearticle that you sent.

The problem of embeddings of S^(n-1) into S^n is a difficult one with a long history. The case of n=2 was essentially solved by Schoenflies. He proved that if C is a simple closed curve in the plane then there is a homeomorphism of the plane to itself carrying C to the unit circle. (Adding the point at infinity doesn't materially change the problem.) In other words, knot theory in this case is trivial.

This is false in higher dimensions. The Alexander Horned Sphere is a famous counterexample; it gives an S^2 embedded in S^3 with interior not homeomorphic to the ball B^3. To try to fix this, one can require that the embedding be "locally flat", meaning you can extend the embedding to a thickened sphere. Then the question becomes the Schoenflies conjecture: such an embedding can be extended to a homeomorphism of S^n. This was solved by Morton Brown and Barry Mazur; see

Brown, Morton (1960), A proof of the generalized Schoenflies theorem. Bull. Amer. Math. Soc., vol. 66, pp. 74–76.

If one restricts embeddings to smooth embeddings, then I believe the problem of unknotting remains open in the case n=3. In this case, we are asking to extend to a diffeomorphism of S^4.

@Prof.  Singer

Thank you so much Prof.  Singer for your answer.