Rust certainly has C++ attitude to performance, for example 'zero-cost abstractions'. That precludes GC. But this is more indirect influence: targeting the same niche
This isn't the case, there are tracing garbage collectors implemented as libraries [1][2], and there is consideration being made for supporting tracing GC in the language and stdlib [0]. As well as reference counted GC having been available in stdlib for a long time [3] (similar to C++'s std::shared_ptr)
[0] https://github.com/rust-lang/rfcs/blob/master/text/1398-kind...
[1] https://manishearth.github.io/blog/2016/08/18/gc-support-in-...