Short answer: No, mapping your domain to your TSB website should not negatively affect your SEO.

Longer technical answer: Domain mapping to TSB is handled via an “A record”….which means it’s technically not a redirection at all (like a “cname” would be) you’re actually pointing your domain to the IP address of the TSB servers, just as you would if you were to point your domain name to any other hosting provider.

Another thing to keep in mind is if you had a previous website under that domain.  Because your new TSB website will most likely have different content, and definitely a different layout and structure, than your previous website, you may experience temporary fluctuations in your search engine rankings as Google and others try to sort out your new content.  Eventually, in most cases, your rankings will return to where they were and more often than not will be better than before.

However, Search Engines are mystical creatures, and we can’t predict how they’ll behave, so we aren’t able to make any guarantees about your search engine rankings while using TSB.