What's the correct syntax to define an activity in manifest file

dot means your package name. It's more short type of declaration.

If you define a subclass, as you almost always would for the component classes (Activity, Service, BroadcastReceiver, and ContentProvider), the subclass is declared through a name attribute. The name must include the full package designation. For example, an Service subclass might be declared as follows:

<manifest . . . >
     <application . . . >
         <service android:name="com.example.project.SecretService" . . . >
             . . .
         </service>
         . . .
     </application> 
</manifest>

However, as a shorthand, if the first character of the string is a period, the string is appended to the application's package name (as specified by the element's package attribute). The following assignment is the same as the one above:

<manifest package="com.example.project" . . . >
     <application . . . >
         <service android:name=".SecretService" . . . >
             . . .
         </service>
         . . .
     </application> 
</manifest> 

When starting a component, Android creates an instance of the named subclass. If a subclass isn't specified, it creates an instance of the base class.

http://developer.android.com/guide/topics/manifest/manifest-intro.html Declaring class names

yes putting the dot is right way.. if you see the eclipse self generated activity it looks like.

 <activity 
        android:name=".MyFirstActivity" 
        android:label="@string/app_name">
    </activity>

so its the right approach, our ide can understand

Using relative paths is fine.
The path is separated by a dot, rather than a slash.

android:name=".one_path_level_down.MainActivity"
android:name=".one_path_level_down.DetailActivity"

The top level is the package level in your manifest file in the "package=". Something like the following:

<manifest xmlns:android="http://schemas.android.com/apk/res/android"
package="com.abc.myapp1" >