olgerm said:
So if first index is up it is 1. kind Christoffel symbol and is first index is down it is 2. kind Christoffel symbol. These are not both same quantities coavariant and contravariant parts because Christoffel symbol is not tensor?
The phrase "covariant and contravariant parts" has no meaning.
Any quantity, (regardless of whether tensor or not), is not the "same" when you raise or lower one of its indices with the metric.
E.g., suppose we're working with an ordinary vector space ##V##. Components of an element ##v \in V## are typically written as ##v^\mu##. If we now work with a tensor product space ##V\otimes V##, we write the components of an element ##h \in V\otimes V## as ##h^{\mu\nu}##.
If we are working with the
dual space of ##V##, denoted ##V^*##, components of an element ##u\in V^*## are usually denoted as ##u_\mu##.
For finite-dimensional vector spaces, the primal and dual spaces ##V## and ##V^*## are isomorphic -- we can find a 1-to-1 mapping between them. In relativity, this is usually done with the help of a metric tensor ##g \in V^* \otimes V^*##, whose components are denoted ##g_{\mu\nu}## and its inverse ##g^{\mu\nu} \in V \otimes V## (defined via ##g_{\mu\alpha} g^{\alpha\nu} = \delta^\nu_\mu##).
Hence our ##v^\mu## (components of an element of ##V##) can be mapped to (components of) an element of ##V^*##, denoted ##v_\alpha##, using the recipe ##v_\alpha = g_{\alpha\beta} v^\beta##.
Once again, for emphasis, elements of ##V## and ##V^*## are not a priori the same in any sense. Rather, we impose a 1-to-1 mapping between them using the metric.
The same ideas apply to
any object with a mixture of upper and lower indices. E.g., something like ##\delta^\nu_\mu## denotes the components of an element of ##V \otimes V^*##.
BTW, if all this stuff about vector spaces, dual spaces, and tensor products, etc, seems like gobbledegook, then you
really need to study a suitable textbook. Since you're obviously studying GR, which textbook(s) are you working from?
