How Can I Echo This PHP 'Field Required' Message On My HTML Page?

I'm building a PHP form. This script cycles through the form's fields and tells me if they're empty or not. When I submit the form, it redirects me from index.html to contact.php and displays a blank page. I would like to display $errmsg above the form in index.html. What am I doing wrong? Here's where I'm pulling information from: http://www.w3schools.com/php/php_form_required.asp

PHP:

<?php
$fields = array('name', 'email', 'telephone');
$error = false;
foreach($fields AS $fieldname) {
        if(!isset($_POST[$fieldname]) || empty($_POST[$fieldname])) {
                $errmsg = 'Please Fill All Fields. '.$fieldname.' is empty!<br />';
                $error = true;
    }
}

HTML:

<form class="form-horizontal" role="form" method="post" action="contact.php">
    <div class="form-group">
            <span class="error">* <?php echo $errmsg;?></span>

Here is my entire contact.php in case the issue is further down.

<?php
$fields = array('name', 'email', 'telephone');
$error = false;
foreach($fields AS $fieldname) {
        if(!isset($_POST[$fieldname]) || empty($_POST[$fieldname])) {
                $errmsg = 'Please Fill All Fields. '.$fieldname.' is empty!<br />';
                $error = true;
        }
}

if(!$error) {
        if (isset($_POST["submit"])) {
                $name = $_POST['name'];
                $email = $_POST['email'];
                $telephone = $_POST['telephone'];
                $message = $_POST['message'];
                $headers = "From: [email protected]\r\n";
                $to = '[email protected]';
                $subject = 'New Lead';
                $body = "From: $name\n E-Mail: $email\n Phone: $telephone\n Message: $message";
        mail('[email protected]', $subject, $body, $headers);
        header("Location: http://www.example.com/messagesent.html");
        die();
        }
}
?>

edit: Entire HTML Form. It's nested in Bootstrap sorry for spaghetti

<!-- Contact Form -->
<form class="form-horizontal" role="form" method="post" action="contact.php">
        <div class="form-group">
                <span class="error">* <?php echo $errmsg;?></span>
                <label for="name" class="col-sm-2 control-label">Name</label>
                <div class="col-sm-10">
                        <input type="text" class="form-control" id="name" name="name" placeholder="First & Last Name">
                </div>
        </div>
        <div class="form-group">
                <label for="email" class="col-sm-2 control-label">Email</label>
                <div class="col-sm-10">
                        <input type="email" class="form-control" id="email" name="email" placeholder="[email protected]">
                </div>
        </div>
                <div class="form-group">
                <label for="telephone" class="col-sm-2 control-label">Phone</label>
                <div class="col-sm-10">
                        <input type="text" class="form-control" id="telephone" name="telephone" placeholder="250-555-555">
        </div>
        </div>
        <div class="form-group">
            <label for="message" class="col-sm-2 control-label">Message</label>
            <div class="col-sm-10">
                    <textarea class="form-control" rows="4" name="message"></textarea>
            </div>
    </div>
    <div class="form-group">
            <div class="col-sm-10 col-sm-offset-2">
                    <input id="submit" name="submit" type="submit" value="Send" class="btn btn-primary">
            </div>
        </div>
        </div>
</form>
<!-- End Contact Form -->

Solution

**Use Ajax Method**    

**index.html** 

    <!DOCTYPE html>
    <html lang="en">
    <head>
    <meta charset="utf-8">
    <title>using Ajax in jQuery</title>
    <style type="text/css">
        label{
            display: block;
            margin-bottom: 10px;
        }
    </style>
    <script src="https://code.jquery.com/jquery-1.11.3.min.js"></script>
    <script type="text/javascript">
    $(document).ready(function(){
        $("form").submit(function(event){
            // Stop form from submitting normally
            event.preventDefault();

            /* Serialize the submitted form control values to be sent to the web server with the request */
            var formValues = $(this).serialize();

            // Send the form data using post
            $.post("contact.php", formValues, function(data){
                // Display the returned data in browser
                $("#result").html(data);
            });
        });
    });
    </script>
    </head>
    <body>
        <form>
            <label>Name: <input type="text" name="name"></label> 
            <input type="submit" value="Send">
        </form>
        <div id="result"></div>
    </body>
    </html>             



contact.php 


<?php
$fields = array('name');
$error = false;
foreach($fields AS $fieldname) {
        if(!isset($_POST['name']) || empty($_POST['name'])) {
                $errmsg = 'Please Fill All Fields. '.$fieldname.' is empty!<br />';
                $error = true;
        }
}

if($error==true){
    echo $errmsg;
    die();
}
else{
        if (isset($_POST["submit"])) {
                $name = $_POST['name']; 
                $headers = "From: [email protected]\r\n";
                $to = '[email protected]';
                $subject = 'New Lead';
                $body = "From: $name\n E-Mail: $email\n Phone: $telephone\n Message: $message";
        mail('[email protected]', $subject, $body, $headers);
        header("Location: http://www.example.com/messagesent.html");
        die();
        }
}
?>