# What are necessary & sufficient conditions for exponential family representation to have complete statistic $T(X)$?

My textbook gives the following theorem for exponential families:

Let $$X_1, \dots, X_n$$ be a random sample from an exponential family with pmf/pdf of the form $$f(x|\theta) = h(x) c(\theta) \exp (w(\theta)t(x))$$ Then the statistic $$T(X) = \sum_{i=1}^n t(X_i)$$ is complete.

But I'm unsure how to interpret this. It does not mention whether the family is curved, full rank, or of minimal dimension. And I feel like these conditions will certainly affect whether the statistic $$T(X)$$ is complete or not.

Am I right, and that extra conditions are not explicitly stated? Or is simply being an exponential family enough to imply completeness of $$T(X)$$?