When can "that" be replaced with an -ing form?

Is there a general rule that allows to replace "noun + that + verb" with "noun + ing-form of verb"?

Consider this sentence:

Metal physical properties are key traits that determine the quality of products.

Can this be written as:

Metal physical properties are key traits determining the quality of products.

The second sentence already has a main verb and key traits is it an object? I am confused.

I would like to know in general when can "that" be omitted? If this question has already been asked I can't find it through the search.

Solution 1:

[1] Metal physical properties are key traits [that determine the quality of products].

[2] Metal physical properties are key traits [determining the quality of products].

It's not a matter of omitting "that" in your second example. Though the bracketed clauses in [1] and [2] are semantically similar, they belong to different categories, the former a relative clause, the latter an ing participial clause.

In [2] "key traits determining the quality of products" is a noun phrase functioning not as object but as predicative complement of "be". The head word is "traits" and the participial clause is its modifier.