Is it possible to infringe a patent by infringing dependent claims and not independent claims? Does that mean I just need to check independent claims of a patent to be sure I don't infringe the patent? Does this apply to US, Europe and other countries?
In theory, you do not infringe a patent based on the scope of a dependent claim if you do not infringe the patent based on the scope of a respective independent claim.
In practice, it is possible to find granted patents in which the dependent claims undo a feature of a claim it depends upon and adds a different feature, they are sometimes called false dependent claims. In that case, you might infringe the patent based on the scope of a dependent claim but not of its independent claim.
Sometimes the patent law does not correctly specify that the dependent claims include all features of the claim they depend upon. The defendant could argue that, in theory, a dependent claim shall include all the features of the claim it depends upon, but what the outcome of that may be is unclear. It is not reassuring that the dependent claim, by definition, shall be more limited than the independent claim when you see granted patents with that type of claims: they are granted and they are problematic due to the uncertainty they produce. Reviewing the dependent claims to check if there are no false dependent claims is advisable.
If you mean that your art mismatches the independent claim on which the dependent claim overtly calls out as its independent claim, then under all WIPO-signatory nations (and certainly the USA), your art does not infringe the dependent claim because it does not infringe the independent claim as well. Each dependent claim is an implied logical-AND extension of the independent claim that the dependent claim overtly calls out as its basis. Your work does not satisfy the logical-AND there, hence no possibility of infringement.
This assumes that the court & jury (e.g., in the USA) agree with your self conclusion that your work mismatches the independent claim on which the dependent claim depends. Perhaps you are over-confident due to yearning for a particular outcome. To forestall that overconfidence, you should consult with a patent attorney to see if the patent attorney likewise as a rather neutral party also agrees that your work is a total mismatch to the metes & bounds carved out by the independent claim that is called out by the dependent claim that partially matches your work.
By definition, in all countries, dependent claims are a narrowing of independent claims. If you do not infringe an independent claim you cannot infringe a claim that depends from it. Period.
However, an independent claim might be hard to understand and apply to the alleged infringing product. A dependent claim might add limitations that make it more clear what the invention is and therefore make a jury see more clearly that you infringe.