Instead of just looking for the Theory of Elasticity and Differential Geometry, I suggest that you look for (Continuum Mechanics and Differential Geometry).
The Theory of Elasticity is included in Continuum Mechanics so that you will find something close to your research area.
Please try to check the following books:
1- "Differential Geometry Applied to Continuum Mechanics", 2009, by Daniel Aubram.
2- "The Geometrical Language of Continuum Mechanics ", 2010, by Marcelo Epstein.
3-"Tensor Analysis and Elementary Differential Geometry for Physicists
and Engineers", 2017, Hung Nguyen-Scha¨fer • Jan-Philip Schmidt