You have posted that:
-------“There is no non-compete agreement between us and they did not mention any such things in the offer letter too.”
Since you have not signed any such agreement there won’t be any breach…………………….
--------“register the domain through my brother and his name’
The owner is your brother, and not you.
Article 21 of the Constitution of India guarantees the live to livelihood and since it is a fundamental right it is held to be inviolable. This makes the enforcing of non compete clauses in Indiaeven more of a difficult task.
Employee can not be restrained to join any other employer after separation and non compete clause amounts to restraint of lawful profession.
An agreement, which restrains anyone from carrying on a lawful profession, trade or business, is void to that extent. Under section 27 of the Indian Contract Act, 1872 agreements in restraint of trade are void.
Indian courts have consistently refused to enforce post-termination non-compete clauses in employment contracts, viewing them as "restraint of trade" impermissible under Section 27 of the Indian Contract Act, 1872, and as void and against public policy because of their potential to deprive an individual of his or her fundamental right to earn a livelihood.
Post term restrictive covenants have been held invalid …………………an employer is not entitled to protect himself against competition on the part of an employee after the employment has ceased.
Apparently it seems you have gone thru the offer letter for ‘Non Compete Clause’.
You may once again carefully go thru job advertisement, interview call letter, offer letter, appointment letter, Employment book, corrigendum if any supplied to you or signed by you, emails etc if any………………………………and look for other clauses e.g.; Non Disclosure, Confidential information, Trade Secrets, Database etc……………….
The ‘Non Disclosure’, ‘Confidential Information’ may stand as an exception and, its validity may be upheld in post employment period.
There are many threads on similar subjects which you may find interesting e.g:
https://www.lawyersclubindia.com/forum/Legal-issues-in-joining-client-company-85262.asp#.UffhMNKAqWM
It shall be appropriate to show your documents to a competent and experienced labor consultant/service lawyer/law firm, specializing in such matters, give inputs in person…………………….
The lawyer that has seen all documents and has analyzed the merits can advise you the best, and let your lawyer’s opinion be final.