Air in jars after canning process

This is a great question. I'm always extra cautious when it comes to canning procedures, following the canning jar companies' instructions to the letter, and relying only on recipes and instructions/methods from reliable sources - a limited pool, in my opinion.

Whenever I have a question, I go to trusted reference sites - the USDA being one of them, as well as any state agricultural extension service. I often consult the Ball Jar Company's "Blue Book" on preserving.

Here is a useful explanation on headspace from an excellent resource for canning - quite a comprehensive knowledge base - from the National Center for Home Food Preservation, a USDA resource hosted by the University of Georgia, found at https://nchfp.uga.edu/questions/FAQ_canning.html#4 :

Do I really need to leave a certain amount of headspace in the jar?
Yes, leaving the specified amount of headspace in a jar is important to assure a vacuum seal. If too little headspace is allowed the food may expand and bubble out when air is being forced out from under the lid during processing. The bubbling food may leave a deposit on the rim of the jar or the seal of the lid and prevent the jar from sealing properly. If too much headspace is allowed, the food at the top is likely to discolor. Also, the jar may not seal properly because there will not be enough processing time to drive all the air out of the jar.

Here's another helpful resource, from the Colorado State Extension site (a resource I rely on a lot for all kinds of things, as I live in Colorado):

https://extension.colostate.edu/topic-areas/nutrition-food-safety-health/canning-tomatoes-and-tomato-products-9-341/

You'll find in that article a link to a variety of USDA publications on canning at http://nchfp.uga.edu/publications/publications_usda.html

Bookmark these resources. You'll be glad you did. ;o)

I'm an old canner of many years. If those are the pictures of your jars before undergoing the canning process, then they were very over-filled. You must leave sufficient headspace in the jar to allow for expansion during the process, which for tomatoes is usually 1/2 inch measured from the top of the jar. If you use a pressure canner, the increased pressure inside during the process causes the contents of your jar to expand to force the air out. As the jar cools, and the contents adjust to normal room pressure, the contents contract and suck down the lid- creating the vacuum seal. The reason you have increased area inside the jar now is because some of the contents were forced from the jar during that process.
You were fortunate that your jars did in fact seal - that is, if they really did. You need to check that, a little closer than the usual pressing on the lid or looking at the lid button. Remove the rings, and clean well around the edge of the lid and thread portion of the jar. Then lift the jar by the edge of the lid, so you hold the weight of the jar by the edge. If the jar is correctly sealed, the lid will stay in place. If it isn't then obviously, it's going to come off. If done within a day of initial processing, unsealed jars can be cleaned, contents removed down to the correct headspace level, and resealed with a brand new lid- or chilled and eaten quickly. Or frozen for later use.
Be careful to keep those jars in a nice dry spot without the rings on them. Periodically check the lids to be sure they remain sealed- as it is possible to lose a seal that happens after overflow in processing. Make sure when you open the jar that you get that "pop" and the sound of air rushing into the jar. And in the future, pay close attention to maintaining the headspace between jar lid and contents, because it is an important detail.

I'm not a canner, but I did some research for you. According to this website: https://hgic.clemson.edu/factsheet/most-frequently-asked-canning-questions/

"Do I really need to leave a certain amount of headspace in the jar? Yes, leaving the specified amount of headspace in a jar is important to assure a vacuum seal. If too little headspace is allowed the food may expand and bubble out when air is being forced out from under the lid during processing. The bubbling food may leave a deposit on the rim of the jar or the seal of the lid and prevent the jar from sealing properly. If too much headspace is allowed, the food at the top is likely to discolor. Also, the jar may not seal properly because there will not be enough processing time to drive all the air out of the jar."

Also "Should liquid lost during processing be replaced? No. Loss of liquid does not cause food to spoil, though the food above the liquid may darken. If, however, the loss is excessive (for example, if at least half of the liquid is lost), refrigerate the jar(s) and use within two to three days."

I guess the question is, did your jars seal? And did you lose half the liquid? If the jars did not seal properly, or if there is liquid loss of half the jar size or more, you should probably refrigerate them and eat them right away.