Thanks for the insight everyone. I am ready to go another round with this thing...
MartinR: I think I have all of the relevant curl packages installed. Maybe I have less due to my particular installation...
# yum list | grep curl
curl.x86_64 7.19.7-40.el6_6.4 @rhel-6-server-rpms
libcurl.x86_64 7.19.7-40.el6_6.4 @rhel-6-server-rpms
python-pycurl.x86_64 7.19.0-8.el6 @anaconda-RedHatEnterpriseLinux-201409260744.x86_64/6.6
libcurl.i686 7.19.7-40.el6_6.4 rhel-6-server-rpms
libcurl-devel.i686 7.19.7-40.el6_6.4 rhel-6-server-rpms
libcurl-devel.x86_64 7.19.7-40.el6_6.4 rhel-6-server-rpms
# yum list installed | grep curl
curl.x86_64 7.19.7-40.el6_6.4 @rhel-6-server-rpms
libcurl.x86_64 7.19.7-40.el6_6.4 @rhel-6-server-rpms
python-pycurl.x86_64 7.19.0-8.el6 @anaconda-RedHatEnterpriseLinux-201409260744.x86_64/6.6
Turner: I was not able to get the logto path code working. I did some experimenting, and maybe with a bit more I could get it to work. Nothing yet though.
Guest: I looked around for the debug mode, and discovered the setting...
conf/dokuwiki.php:$conf['allowdebug'] = 0;
I think we are in business. After enabling it and going through the motions, here is what I discovered...
# tail -100 dokuwiki/data/cache/debug.log
07:57:17 10.13.14.35: checkUpdateMessages(): downloading messages to /wikiprdap01data/dokuwiki/data/cache/f/f5c8f522bea56a6958a8556e463ed34a.updmsg
07:57:17 10.13.14.35: checkUpdateMessages(): unexpected HTTP response received
07:57:25 10.13.14.35: checkUpdateMessages(): messages up to date
07:57:27 10.13.14.35: checkUpdateMessages(): messages up to date
07:57:31 10.13.14.35: checkUpdateMessages(): messages up to date
07:57:32 10.13.14.35: checkUpdateMessages(): messages up to date
And the file mentioned above was empty. We are getting closer!!
Also, according to this article...
https://www.dokuwiki.org/devel:debugging I can append "&do=debug" or "&do=check" to the URL and get additional information.
do=check told me some information, but it seemed irrelevant to this problem. do=debug gave me a ton of info about my particular installation, but not directly about this specific error.