ABSTRACT
Metabolic and hyperbolic spaces will be defined in terms of direct orthogonal sums of isotropic planes. Accordingly we begin with a complete description of isotropic planes depending on characteristic of the ground field. Then we explain the notion of direct orthogonal sum of bilinear spaces and proceed to hyperbolic and metabolic spaces. These are fundamental objects in the construction of the Witt ring of symmetric bilinear spaces over a field. When the characteristic of the ground field is different from two there is no need to distinguish between hyperbolic and metabolic spaces and the entire theory is much simpler. However, we intend to introduce Witt rings for all fields, and for this the right concept is metabolic spaces rather than the hyperbolic ones.
