Mystery Theorem

[Qdogsman]

Please read and consider the following open letter I recently sent to an old professor of mine:

 

Dear Jerry,

 

After talking with you yesterday I decided to write this as an open letter to the world. Thank you for your insight; it was good to hear from you again after 45 years.

 

You asked me to be more specific about my question of the existence of a certain theorem, and you suggested that a mathematician who is more knowledgeable in Differential Geometry than you might be better able to help. So I want to open this letter up and get it in front of as many of those mathematicians as possible.

 

Then, to be fair, I thought I should tell you why I am so interested in this "mystery theorem". In thinking about how to do that, I realized that this letter may hold interest for a much wider audience. As a result, I am writing it as an open letter to the world.

 

Without getting into my personal beliefs, I happen to be among those people who believe that there is more to existence than the observable and accessible 4D space-time of science. This puts me squarely in the category of virtually all religious believers, including those who believe in parapsychology, animism, spiritualism, or of other believers in the supernatural of various flavors.

 

The rest of the people, typically including most scientists, believe that our 4D world is all that exists.

 

Since a definitive answer to my question could shed some light on the controversies swirling around the divide between these two categories of belief systems, a search for an answer might be interesting to anyone involved in the debate.

 

In the 1990's I distinctly remembered you proving a "theorem", back in the 1960's, in Differential Geometry that you summed up for the class afterward in a sentence that was very close to, "There you see? You can't have a bent space unless it is an embedded manifold in a space of at least one dimension higher than that of the bent space."

 

I also distinctly remember the impact that result had on me. I at once realized that if our 4D space-time were "bent", then the necessary additional dimension would provide the "space" for God, or angels, or all sorts of other extant but inaccessible beings, places, and structures to exist.

 

During most of my adult life, I wondered how this result could shed light on the profound questions related to the meaning of life. More than that, I wondered why nobody seemed to be taking advantage of the theorem and applying it.

 

Then in the late 90's, I had an experience that shook my faith to the core—my faith, that is, in my own memory. In a discussion on an Internet forum relating to the rift between science and religion, I confidently expressed my belief in the "mystery" theorem. When challenged for a reference, I pulled out my copy of Michael Spivak's Calculus on Manifolds, which I thought you had used in class, and looked for the "theorem". To my shock and consternation, the "theorem" simply was not there.

 

To this day, I can't explain where my memory failed me. You have straightened me out on the question of whether it was you I was listening to. You said it was not. You said you taught Differential Geometry but not at the location where I took the course. My memory of the reference also clearly failed because the "theorem" is nowhere in Spivak's book.

 

Since that classroom experience had such a life-altering impact on me, it is clear that either some other professor, in some other class, proved the "theorem", or that the memory got planted by a dream or in some other mysterious manner. Getting a definitive answer to my question will help put my mind at ease. Please, someone help me.

 

Even though I seem to remember the professor using the term 'bent', I realize that 'bent' is not a well-defined mathematical term. I also realize that the terms 'curve' and 'curvature', while they are well-defined, are not simple concepts. So I am at a loss to answer your first question in response to me: you asked me, "What do you mean by 'bent'?"

 

I don't know.

 

But it is easy to think of an intuitive example of how the "theorem" should hold in some specific cases. For example, you can't "bend" a sheet of 2D paper that is lying on a tabletop without "bending" part of it into the third dimension of the room in which the table is located. The question is, can you "bend" any space without having an extra dimension "in which to bend it"?

 

Assuming that there is an appropriate definition of 'curvature' such that the "mystery theorem" holds, the theorem would say:

 

At least n+1 dimensions are necessary for a particular curvature to exist in an n-dimensional space, and the n-space must be an embedded manifold in the (n+1)-space.

 

If such a theorem exists, it would seem to apply to the current controversy between science and religion. Since from General Relativity we know that our 4D space-time is curved, or "bent" in the presence of energy, if the theorem is true, and if that particular curvature qualifies, then it would imply that a higher-dimensional world exists in which ours is an embedded manifold.

 

The extra dimension(s) would be inaccessible to us and to any of our instruments because everything in our 4D world is made of 3D stuff moving in a 1D time-like direction. Therefore there should be no mystery as to why we don't "see" those extra dimensions, which commonly seems to be the reason scientists don't take the possibility seriously.

 

It seems to me that String Theory, for example, could be greatly simplified if the extra dimensions didn't need to be "curled up" in order to keep them out of sight. Maybe Theodor Kaluza's suggestion to Einstein should be resurrected and pursued without encumbering it with Oskar Klein's compactification. It seems to me that the search for the magic Calabi-Yau space could be shifted from the haystack to the well-lighted ground under the proverbial lamppost.

 

I sincerely hope that someone who can definitively answer my question will somehow provide the answer to me. If that is not you, I beg you to forward this to someone who might understand the question and who might be able to cite the theorem, prove it to be false, or begin working on finding a proof one way or the other.

 

Sincerely,

 

Paul R. Martin

Seattle, WA USA