According to my lecture notes on relativity the variation of a Christoffel symbol $\delta\Gamma^\gamma_{\alpha\beta}$ is a tensor, in contrast to $\Gamma^\gamma_{\alpha\beta}$ itself.
I don't see how to arrive at such a result.
Also, if the variation of $\Gamma^\gamma_{\alpha\beta}$ can make $\Gamma^\gamma_{\alpha\beta}$ behave like a tensor, then is it also possible that the variation of a tensor is not a tensor?
Best Answer
You can verify yourself that the difference between any two connections (entities that change according to the same law as the Christoffel symbols when changing coordinates) with the same indices is a tensor by seeing how it behaves with respect to coordinate change. The non-tensor terms cancel. This is also true for two connections that are very close to another in value, such as the variation in a single Christoffel symbol.
The answer to your second question is "no". Again, you can verify yourself that the difference between two tensors is a tensor. I would also like to give an example of a different question along the same line of thought: There are non-rational real numbers whose difference is rational (for instance $\pi$ and $4+\pi$). Does that mean that the difference between two rational numbers could be non-rational?
Note: I'm not using that example to tell you that your question is stupid; it's not. Asking these kinds of questions helps you understand what a tensor is. The main takeaway from my answer should be "a tensor is a mathematical object whose components with respect to a given coordinate system changes according to a specific law when changing to another coordinate system." This is the first and foremost way of showing that something is or isn't a tensor, and only in certain cases are there better / faster ways. One of those ways is that sums, differences, products and contractions of tensors are tensors.