Recent developments in robotics increasingly highlight the importance of sensing technology, especially tactile perception, in enabling robots to effectively engage with their environment and interpret physical interactions. Due to power efficiency and low cost, the triboelectric mechanism has been frequently studied for measuring pressure and identifying materials to enhance robot perception. Nevertheless, there has been limited exploration of using the triboelectric effect to detect curved surfaces, despite their prevalence in daily lives. Here, a triboelectric multimodal tactile sensor (TMTS) of multilayered structural design is proposed to recognize distinct materials, curvatures, and pressure simultaneously, thus decoupling different modalities to enable more accurate detection. By attaching sensors to robotic fingertips and leveraging deep learning analytics, the quantitative curvature measurement provides more precise insights into an object's detailed geometric characteristics rather than merely assessing its overall shape, hence achieving automatic recognition of 12 grasped objects with 99.2% accuracy. The sensor can be further used to accurately recognize the softness of objects under different touch gestures of a robotic hand, achieving a 94.1% accuracy, demonstrating its significant potential for wide-ranging applications in a future robotic-enabled intelligent society.