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The aim of the present paper is to study a Hofer-like geometry on the space of Legendrian submanifolds of a contact manifold. Using contact Hofer energy we define a pseudometric on this space and find sufficient conditions that this be nondegenerate. It is proved that an ample class of distances reduces to multiples of that metric. Examples of subsets with positive contact displacement energy are given.

Hofer type geometry on the space of Legendrian submanifolds of a contact manifold