There's nothing wrong with those sentences. In fact, the second one -"I saw the man who lives next door" - has to be in the present tense if you're talking about your current next-door neighbour. If you said "I saw the man who lived next door" this would mean that he doesn't live there any more. The other two are also OK in the present : her name is still Anne and he still likes to read. You might use a past tense in this situation. "What's that girl called? She said her name was Anne, but every seems to call her Lucy." There is no general rule about subordinate clauses having to be in the past tense if the main clause is past. The only rule of this type is the one applying to indirect speech: the words that are reported go one step back into the past: present becomes past, past becomes past perfect, 'will' becomes 'would', and so on. However, this 'one step back' process isn't strictly necessary if what the person said still holds true at the present time. This is certainly the case with universal truths (such as your sun example), but also with more banal situations. For example, if you're relaying a message that you've just received, or repeating someone's mumbled comments to someone who's hard of hearing, you wouldn't do the one step back process - "Grandma wants to know if you want sugar in your tea, Aunt Agatha!". Or compare these two examples. 1. You've booked a taxi and the taxi company sends you a message. You might report this to your friends by saying "They said the taxi's on its way and it'll be here in a minute". Here, you're using a present and a future tense in the subordinate clause because this reflects the relevance of the statement to the present moment. 2. You wait for ten minutes and the taxi doesn't appear. At this point you might report the content of the message again by saying "They said it was on its way and it would be here in a minute". The tense has shifted because it's no longer real time. I hope that makes sense.

To me, as a user of English, rather than a teacher, I would say that we use past tense when we want to indicate that what was said was only true at the time, and we don't want to indicate that we have any reason to suggest that it is true now. We use present to indicate that whatever was said is known, or can reasonably be expected to still be true now. Her name was Anne now, and there is no reason to expect that to change. If the speaker used 'lives', it means he lived there then, and still does now. To me, the reading one could be said either way. I don't think grammatically - I (hopefully) know what is right intuitively. I only learnt grammar when I learnt other languages. I'm still learning it to answer questions here ;)

Мне тоже стало любопытно, и я посмотрел ссылки в Википедии. Вывод: в середине века (не путать со "в средние века":)) англичан учили в школе, что так говорить неправильно, кроме случая universal truth. Они, тем не менее, говорили. С тех пор, кажется (но я не смотрел школьные учебники), это всё отменилось и в современной редакции последовательность необязательна, если речь о продолжающейся ситуации. Вот что у меня открыто в браузере: 1. Webster's Dictionary of English Usage: English Dictionary https://books.google.ru/books?id=opdWDwAAQBAJ&pg=PA838 он же: https://list-english.ru/img/newpdf/dictpdf/6.pdf Здесь есть обзор предшествующей литературы. 2. English Grammar: A University Course http://toefl.uobabylon.edu.iq/papers/itp_2015_1817487.pdf#page=326 Учебник с функциональной перспективы. Ваш вопрос даётся в контексте связанных явлений: не только is становится was но и this норовит стать that. 3. https://archive.org/details/understandinggra013304mbp/page/n181 Understanding Grammar, Paul Roberts, 1954. Это ссылка из dictionary of English grammar. Прикольная книжка.

Thanks Fernando, I didn't know the rule about verbs going one step back in the past applies (applied???) to indirect speech only. Good to know. Thanks Hasib, I'm not sure I understand the first explanation (I guess I haven't read enough grammar books because I honestly have no idea what "inactive verb" could possibly mean), but the second one makes sense to me. And still, they can say "He didn't know where Mongolia WAS" and no one would blink an eye, despite Mongolia not changing its location! Why is that?

1.She said her name is Anne (as we know to make one sentence only need one or more subject but one principle verb,in this sentence said is not main verb, SAID is here used as verb-3.Also we know that before verb-3 if no auxiliary verb are used that verb remain inactive verb.So is the main verb. Also you can think other way,our name never change as what kept when we birth.So, if we use was that means in past time her name was Anne,now changed to another name. I THINK YOU UNDERSTAND.