Redirect www to non-www keeping the path

Question

In my .htaccess, the redirect from "www.example.com" to "example.com" works.
But when the urls is "www.example.com/someotherpath.html" the redirect is still "example.com"

How to solve this?

I am working with this settings in .htaccess:

<IfModule mod_rewrite.c>

############################################
## enable rewrites

    Options +FollowSymLinks
    RewriteEngine on

############################################
## you can put here your magento root folder
## path relative to web root
RewriteBase /
#########################
# redirect www to no-www
#########################

RewriteCond %{HTTP_HOST} ^www\.(.+) [NC]
RewriteRule ^(.*) http://%1/$1 [R=301,NE,L]
    #RewriteBase /magento/

############################################
## uncomment next line to enable light API calls processing

#    RewriteRule ^api/([a-z][0-9a-z_]+)/?$ api.php?type=$1 [QSA,L]

############################################
## rewrite API2 calls to api.php (by now it is REST only)

    RewriteRule ^api/rest api.php?type=rest [QSA,L]

############################################
## workaround for HTTP authorization
## in CGI environment

    RewriteRule .* - [E=HTTP_AUTHORIZATION:%{HTTP:Authorization}]

############################################
## TRACE and TRACK HTTP methods disabled to prevent XSS attacks

    RewriteCond %{REQUEST_METHOD} ^TRAC[EK]
    RewriteRule .* - [L,R=405]

############################################
## redirect for mobile user agents

    #RewriteCond %{REQUEST_URI} !^/mobiledirectoryhere/.*$
    #RewriteCond %{HTTP_USER_AGENT} "android|blackberry|ipad|iphone|ipod|iemobile|opera mobile|palmos|webos|googlebot-mobile" [NC]
    #RewriteRule ^(.*)$ /mobiledirectoryhere/ [L,R=302]

############################################
## always send 404 on missing files in these folders

    RewriteCond %{REQUEST_URI} !^/(media|skin|js)/

############################################
## never rewrite for existing files, directories and links

    RewriteCond %{REQUEST_FILENAME} !-f
    RewriteCond %{REQUEST_FILENAME} !-d
    RewriteCond %{REQUEST_FILENAME} !-l

############################################
## rewrite everything else to index.php

    RewriteRule .* index.php [L]


</IfModule>

Answer 1

Pop this at the top of your .htaccess

RewriteCond %{HTTP_HOST} ^www\. [NC]
RewriteRule ^(.*)$ http://%{HTTP_HOST}%{REQUEST_URI} [R=301,L]

Answer 2

With code @Ben Lessani, if you need without www.

RewriteCond %{HTTP_HOST} ^www\. [NC]
RewriteRule ^(.*)$ http://domain.com%{REQUEST_URI} [R=301,L]