FOREX-Dollar on defensive as investors stick with risk, for now

* Dollar index at lowest in over 2 months

* U.S. manufacturing data shows modest rebound

* Australian dollar unchanged after RBA, near highest since Jan

* Graphic: World FX rates in 2020 https://tmsnrt.rs/2RBWI5E

By Hideyuki Sano

TOKYO, June 2 (Reuters) - The dollar was on the back foot on Tuesday as investors maintained their hope in a global economic recovery, despite growing concerns over U.S.-China tensions and mass protests across America over the death of a black man in police custody.

The U.S. dollar index against a basket of six major currencies hovered near its weakest level since mid-March, standing at 97.885.

It has fallen about 5% from a peak hit in March when panic over the COVID-19 pandemic gripped the world's financial markets, prompting investors to scramble for the safety of dollar cash.

"There are some potential flashpoints such as U.S. demonstrations and China-U.S. tensions," said Kyosuke Suzuki, director of forex at Societe Generale. "But, on the whole, the market is still moderately risk-on."

The euro fetched $1.1126, little changed so far on Tuesday but holding near a 2-1/2-month high of $1.1154 touched on Monday.

The common currency gained traction after the European Union's executive last month unveiled a 750 billion euro plan to prop up economies hammered by the coronavirus pandemic.

Sterling hit a one-month high of $1.2525 before stepping back to trade flat $1.2479.

U.S. manufacturing activity inched up from an 11-year low in May, an index showed, and although the reading was weaker than forecast, it aligned with market expectations that the worst of the economic downturn has passed.

Against the safe-haven yen, the dollar was at 107.68 yen , stuck in a well-worn range between 106 and 108 over the last several weeks.

Market risk sentiment was hurt only slightly on Monday when Bloomberg reported China had told state-owned firms to halt purchases of soybeans and pork from the United States, raising concerns that the trade deal between the world's two biggest economies could be in jeopardy.

Investors' economic optimism has so far also survived the rising social unrest in the United States where President Donald Trump vowed to deploy thousands of heavily armed soldiers and law enforcement to halt violence in the U.S. capital and other cities if mayors and governors failed to regain control of the streets.

The protests erupted over the death of George Floyd, a 46-year-old African-American who died in Minneapolis police custody after being pinned beneath a white officer's knee for nearly nine minutes.

The Australian dollar, often seen as a proxy bet on the strength of the Chinese economy, fetched $0.6789, having reached its highest levels since late January.

The currency hardly budged after the Reserve Bank of Australia kept its monetary policy on hold as expected.

The Chinese yuan gained slightly to 7.1200 per dollar , pulling further away from an eight-month low of 7.1765 touched last week.

"The Chinese authorities do not appear to intend to guide the yuan weaker now, certainly less so than they were when the yuan fell to 7.18 last September," said Masashi Hashimoto, senior economist at the Institute for International Monetary Affairs in Tokyo.

The yuan hit an 11-year low of 7.1854 per dollar in September when diplomatic tensions intensified over Trump's fourth round tariffs on Chinese products.

MSCI's emerging market currency index also rose to its highest levels since mid-March with the Indonesian rupiah, Thai baht and Mexican peso all rising to more than two-month peaks. (Reporting by Hideyuki Sano Editing by Shri Navaratnam and Sam Holmes)