受随机赋范模在条件风险、条件优化和Banach空间局部理论中的应用启发，继最近关于随机严格凸性和随机一致凸性的工作，本项目对随机赋范模的几何理论展开进一步研究。首先对随机赋范模上的L^{0}-值函数定义Gateaux可微性与Fréchet可微性，给出随机光滑与随机一致光滑的随机赋范模的定义；接着讨论随机光滑性、随机一致光滑性分别与随机严格凸性和随机一致凸性之间的对偶关系；然后，研究随机光滑性、随机一致光滑性分别与随机赋范模生成的抽象L^{p}-空间的经典光滑性和一致光滑性之间的内在联系；最后，对随机赋范模引进L^{0}-端点的概念，建立它与随机赋范模生成的抽象L^{p}-空间中通常端点之间的关系，以期探索它们在著名问题"Krein-Milman性质与Radon-Nikodym性质是否等价"中的应用。

Motivated by the applications of random normed modules to conditional risk measures, conditional optimization and the local theory of Banach spaces, after the work of random strict convexity and random uniform convexity, the project is focused on the further study of geometric theory of random normed modules. First, the notions of Gateaux differentiability and Fréchet differentiability are introduced for L^{0}-valued functions on a random normed module, and smooth and uniformly smooth random normed modules are respectively defined. Next, the dual relations between random smoothness and random strict convexity, random uniform smoothness and random uniform convexity are discussed, respectively. Then, the connections between random smoothness (or random uniform smoothness) of a random normed module and classical smoothness (resp. classical uniform smoothness) of the abstract L^{p}-space derived from the random normed module are studied. Finally, the notion of L^{0}-extreme point is introduced in a random normed module and its relations to ordinary extreme point in the abstract L^{p}-space derived from the random normed module are established in the expectation of their potential applications to the famous open problem "Are Krein-Milman property and Radon-Nikodym property equivalent to each other?".