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    $\begingroup$ This is just a vague memory, but I think there may be applications in fields like robotics, where the robot has various degrees of freedom (angle of this joint, length of this telescoping part, ...), and the space describing these degrees of freedom is a manifold. $\endgroup$ Commented Aug 4, 2015 at 19:18
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    $\begingroup$ Here is one reference to support @BenCrowell's remark: Ebetiuc, Silvia, and Staab Haraldw. "Applying differential geometry to kinematic modeling in mobile robotics." In 2005 WSEAS Int. Conf. on Dynamical systems and control, pp. 106-112. 2005. (PDF download link.) $\endgroup$ Commented Aug 4, 2015 at 20:40
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    $\begingroup$ It's used quite heavily in some branches of continuum mechanics. $\endgroup$ Commented Aug 13, 2015 at 21:25