SHANGHAI, Sept 26 (Reuters) - Chinese and Hong Kong stocks fell on Tuesday as lingering economic worries and geopolitical tensions weighed on sentiment, with thin trading seen ahead of China's National Day holiday.

** China's blue-chip CSI 300 Index ended the session down 0.6%, while the Shanghai Composite Index declined 0.4%. Hong Kong's Hang Seng Index lost 0.9% to a 10-month closing low.

** Most sectors dropped in China. The banking sector slid 0.4% and the Defence Index lost 1.1%.

** The recent improvement in some economic activity indices has convinced many market participants that China's economy has already bottomed out, but "we remain cautious," Nomura said in a note.

** Many private developers in lower-tier cities are still being trapped, rise in prices of imported commodities and energy was not passed to downstream sectors, and "geopolitical tensions have not really improved, if they have not deteriorated," Nomura added.

** U.S. President Joe Biden's administration on Monday imposed new trade restrictions on 11 Chinese and five Russian companies, accusing some of supplying components to make drones for Russia's war effort in Ukraine.

** Latest development at China Evergrande and China Oceanwide Holdings reversed a brief early respite for the property sector, which accounts for roughly a quarter of China's economy.

** An index tracking Hong Kong-listed mainland developers dropped 2.2%, as Country Garden and Longfor Group slumped 4.2% and 3.3%, respectively. (Reporting by Shanghai Newsroom; Editing by Rashmi Aich and Varun H K)