How to validate a form without returning a view in asp.net?

The big question as other person have asked is why not use javascript to call an api for login for example with ajax post

$.post("your api here",
{
    userName: "",
    password: "",
    rememberMe: ""
},function(data, status){
     //show user what ever
});

this code is simple enough, But if you cannot use javascript because of anything "I wouldn't know" you will have the below long code and some implementations on your existing controllers

I will explain along

You will have to add a ReturnUrl to your form, change your login form to this

@using Microsoft.AspNet.Identity
@model Rent_a_Car.Models.LoginViewModel
@{
if (Request.IsAuthenticated)
{
    using (Html.BeginForm("LogOff", "Account", new { returnUrl = Request.Url.PathAndQuery }, FormMethod.Post, new { id = "logoutForm", @class = "navbar-right" }))
    {
        @Html.AntiForgeryToken()            
        <ul class="nav navbar-nav navbar-right">
            <div class="dropdown">
                <a class="btn btn-secondary dropdown-toggle" href="#" role="button" id="dropdownMenuLink" data-toggle="dropdown" aria-haspopup="true" aria-expanded="false">
                    @User.Identity.GetUserName()
                </a>
                <div class="dropdown-menu" aria-labelledby="dropdownMenuLink">
                    @Html.ActionLink("My account", "Index", "Manage", routeValues: null, htmlAttributes: new { title = "Manage", @class = "dropdown-item" })
                    <a class="dropdown-item disabled" href="#">My orders</a>
                    <div class="dropdown-divider"></div>
                    <a class="dropdown-item" href="javascript:document.getElementById('logoutForm').submit()">Log off</a>
                </div>
            </div>
        </ul>
    }
}
else
{
    <div class="dropleft">
        <button class="btn btn-secondary dropdown-toggle" type="button" id="dropdownMenuButton" data-toggle="dropdown" aria-haspopup="true" aria-expanded="false"> Log-In </button>
        <div class="dropdown-menu" aria-labelledby="dropdownMenuButton" style="width: 300px">
            @* form starting *@
            @using (Html.BeginForm("Login", "Account", new { ReturnUrl = ViewBag.ReturnUrl }, FormMethod.Post, new { @class = "form-horizontal", role = "form" }))
            {
                @Html.AntiForgeryToken()
                <p>@ViewBag.error</p>@*Display the error here*@
                <div class="col-lg-12">
                    <div class="text-center">
                        <h3><b>Log In</b></h3>
                    </div>
                    <form id="ajax-login-form" method="post" role="form" autocomplete="off">
                        <div class="form-group">
                            <label for="username">Username</label>
                            @Html.TextBoxFor(m => m.Email, new { @class = "form-control", @placeholder = "Email" })
                        </div>
                        <div class="form-group">
                            <label for="password">Password</label>
                            @Html.PasswordFor(m => m.Password, new { @class = "form-control", @placeholder = "Password" })
                        </div>
                        <div class="form-group">
                            <div class="col-xs-5">
                                <input type="submit" name="Login" id="login-submit" tabindex="4" class="form-control btn btn-danger" value="Log In">
                            </div>
                        </div>
                        <div class="col-xs-5" align="Center">
                            @Html.CheckBoxFor(m => m.RememberMe)
                            <label for="remember"> Remember Me</label>
                        </div>
                        <div class="form-group">
                            <div class="row">
                                <div class="col-lg-12">
                                    <div class="text-center">
                                        <a tabindex="5" class="forgot-password">Forgot Password?</a>
                                    </div>
                                </div>
                            </div>
                        </div>
                    </form>
                </div>
            }
            @*form ending*@
        </div>
    </div>
}
@section Scripts {
    @Scripts.Render("~/bundles/jqueryval")
}

Then in your Account Controller, return a redirect to the view where the user is currently, because it is a redirect, you will have to put you model into a TempData which is why @Html.ValidationSummary(true, "", new { @class = "text-danger" }) and some other model error might not work for you, so you do you own validation.

public async Task<ActionResult> Login(LoginViewModel model, string returnUrl)
{
    if (!ModelState.IsValid)
    {
        //here send back the error and the model with validation message to any view
        TempData["LoginModel"] = model;
        TempData["LoginError"] = "Incorrect details";
        return Redirect(returnUrl);
    }
    var result = await SignInManager.PasswordSignInAsync(model.Email, model.Password, model.RememberMe, shouldLockout: false);
    switch (result)
    {
        case SignInStatus.Success:
            return RedirectToLocal(returnUrl);
        case SignInStatus.LockedOut:
            return View("Lockout");
        case SignInStatus.RequiresVerification:
            return RedirectToAction("SendCode", new { ReturnUrl = returnUrl, RememberMe = model.RememberMe });
        case SignInStatus.Failure:
        default:
            TempData["LoginModel"] = model;
            TempData["LoginError"] = "Invalid login attempt";
            return Redirect(returnUrl);                    
    }
}

Now this is where you have alot to do.

You need to go to all the possible controllers where the user might hit first (I will advice all your controller) and get the TempData (model) and error. example the Index method of your CarController can be like:

public ActionResult Index(int Id)
{
    LoginViewModel model = (LoginViewModel) TempData["LoginModel"];
    ViewBag.error = TempData["LoginError"];
    //Do other logic
    return View(model);
}