# mathlibdocumentation

geometry.manifold.algebra.structures

# Smooth structures #

In this file we define smooth structures that build on Lie groups. We prefer using the term smooth instead of Lie mainly because Lie ring has currently another use in mathematics.

@[class]
structure smooth_ring {𝕜 : Type u_1} {H : Type u_2} {E : Type u_3} [normed_group E] [ E] (I : H) (R : Type u_4) [semiring R] [ R] :
Prop

A smooth (semi)ring is a (semi)ring R where addition and multiplication are smooth. If R is a ring, then negation is automatically smooth, as it is multiplication with -1.

Instances of this typeclass
@[protected, instance]
def smooth_ring.to_has_smooth_mul {𝕜 : Type u_1} {H : Type u_2} {E : Type u_3} [normed_group E] [ E] (I : H) (R : Type u_4) [semiring R] [ R] [h : R] :
@[protected, instance]
def smooth_ring.to_lie_add_group {𝕜 : Type u_1} {H : Type u_2} {E : Type u_3} [normed_group E] [ E] (I : H) (R : Type u_4) [ring R] [ R] [ R] :
@[protected, instance]
def field_smooth_ring {𝕜 : Type u_1}  :
theorem topological_semiring_of_smooth {𝕜 : Type u_1} {R : Type u_2} {E : Type u_3} {H : Type u_4} [normed_group E] [ E] [ R] (I : H) [semiring R] [ R] :

A smooth (semi)ring is a topological (semi)ring. This is not an instance for technical reasons, see note [Design choices about smooth algebraic structures].