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We construct a path model for geometric crystals in the sense of Berensteinand Kazhdan. Our model is in every way similar to Littelmann's and tropicalizesto his path model. This paper lays the foundational material for a subsequentwork where we examine the measure induced on geometric crystals by Brownianmotion.  If we call Berenstein and Kazhdan's realization of geometric crystals thegroup picture, we prove that the path model projects onto the group picturethanks to a morphism of crystals that restricts to an isomorphism on connectedcomponents. This projection is in fact the geometric analogue of theRobinson-Schensted correspondence and involves solving a left-invariantdifferential equation on the Borel subgroup.  Moreover, we identify the geometric Pitman transform $\mathcal{T}_{w_0}$introduced by Biane, Bougerol and O'Connell as the transform giving the pathwith highest weight, in the geometric crystal path model. This allows to provea geometric version of Littelmann's independence theorem. The geometricRobinson-Schensted correspondence is detailed in a special section, because ofits importance.  Finally, we exhibit the Kashiwara and Sch\"utzenberger involutions in boththe group picture and the path model.  In an appendix, we explain how the left-invariant flow is related to theimage of the Casimir element in Kostant's Whittaker model.

Littelmann path model for geometric crystals