Nakahar in his book on the Geometry and Topology introduces the Riemannian metric as a type of (0,2) tensor which contains some properties which I interpreted them as a kind of multiplication factor. On the other hand, as we know, the metric is a solution of Einstein equation, hence it describes the gravitational field (maybe it is better to say that it is an auxiliary field since the connection could have a better description of this field on the equivalence principle.). I want to know, how can be connect these two definitions on metric? Why the factor that change the upper indices to lower one and vice versa, is known as a gravitational field?

Nakahara in his book on the Geometry and Topology introduces the pseudo-Riemannian metric as a type of (0,2) tensor which contains some properties which I interpreted them as a kind of multiplication factor. On the other hand, as we know, the metric is a solution of Einstein field equations, hence it describes the gravitational field (maybe it is better to say that it is an auxiliary field since the connection could have a better description of this field on the equivalence principle.). I want to know, how can be connect these two definitions on metric? Why the factor that change the upper indices to lower one and vice versa, is known as a gravitational field?

Nakahar in his book on the Geometry and Topology introduces the Riemannian metric as a type of (0,2) tensor which contains some properties which I interpreted them as a kind of multiplication factor. On the other hand, as we know, the metric is a solution of Einstein equation, hence it describes the gravitational field (maybe it is better to say that it is an auxiliary field since the connection could have a better description of this field on the equivalence principle.). I want to know, how can be connect these two definitions on metric?

Nakahar in his book on the Geometry and Topology introduces the Riemannian metric as a type of (0,2) tensor which contains some properties which I interpreted them as a kind of multiplication factor. On the other hand, as we know, the metric is a solution of Einstein equation, hence it describes the gravitational field (maybe it is better to say that it is an auxiliary field since the connection could have a better description of this field on the equivalence principle.). I want to know, how can be connect these two definitions on metric? Why the factor that change the upper indices to lower one and vice versa, is known as a gravitational field?