echo $ORACLE_HOME
/home/oracle/product/10g/  (We should eliminate the slash / , export ORACLE_HOME=/home/oracle/product/10g) 
It causes error "Connected to an idle instance"

SQL> conn / as sysdba
Connected to an idle instance.

export ORACLE_HOME=/home/oracle/product/10g

SQL> conn / as sysdba
Connected.


Another reason: 

ps -ef|grep pmon

  oracle 19443     1   0   Oct 13 ?          72:50 ora_pmon_dodpp1
  oracle 12421     1   0   Nov 18 ?          27:15 ora_pmon_dodpp5


echo $ORACLE_SID
DODPP1

SQL> conn / as sysdba
Connected to an idle instance.

export ORACLE_SID=dodpp1  (After using lower case, it's ok!)
echo $ORACLE_SID
dodpp1

SQL> conn / as sysdba
Connected to an idle instance.
[@more@]
Connected to an idle instance – while database is running
