  Courant algebroids are a natural generalization of quadratic Lie algebras,
appearing in various contexts in mathematical physics. A connection on a
Courant algebroid gives an analogue of a covariant derivative compatible with a
given fiber-wise metric. Imposing further conditions resembling standard
Levi-Civita connections, one obtains a class of connections whose curvature
tensor in certain cases gives a new geometrical description of equations of
motion of low energy effective action of string theory. Two examples are given.
One is the so called symplectic gravity, the second one is an application to
the the so called heterotic reduction. All necessary definitions, propositions
and theorems are given in a detailed and self-contained way.
