Review by Stephen Parrott 
			of 

		The Geometry of Physics
		   An Introduction 
			by
                Theodore Frankel
		Revised Printing of the First Edition 
		Cambridge University Press, 1997
		(note that this is not the latest edition)

	I will start with the conclusion:  This is a superb book.
I shall have nothing critical to say about it:  only caveats because
no book can satisfy the needs of all readers.

	The only part of this book that I read contiguously and in detail was
the last third entitled "Lie Groups, Bundles, and Chern Forms".
The goal was to firm up my rather informal acquaintance with vector bundles
and related concepts such as connections and their curvatures.  
Back in the early 1960's when I first learned differential geometry 
at the University of Michigan, vector bundles were not even 
in the common mathematical lexicon.  I had never heard of them, 
and I would be surprised to learn a significant number at that place and time 
had either.

	The graduate course that introduced me to differential geometry
would possibly be considered somewhat old-fashioned  by modern (2016) 
standards, though by 1960's standards it may have been 
cutting-edge.  The Cartan viewpoint of "moving frames" 
which is central to Frankel's treatment of these topics was scarcely mentioned.

	Because of this, in order to understand Frankel's treatment of 
vector bundles, I had to frequently go back to study his treatment of 
classical curvature from the Cartan viewpoint.  
Thus I ended up reading most of the book, though not in full detail.

	I am now a convert to the Cartan viewpoint.  I think it offers 
more insight than the way that I originally learned differential geometry,
and I owe this to Frankel's clear exposition.
 
	The whole book seems clearly written, with many 
and generally excellent diagrams.  I found very few typos.
As a book author myself, I can appreciate and marvel at 
the attention to detail and hard work which must 
have gone into each of its 600-odd pages.   Offhand, the only comparable 
book which comes to mind is Misner, Thorne, and Wheeler's "Gravitation".

	I imagine that Frankel would be an excellent introduction 
to differential geometry, though I didn't read the introductory sections 
in detail. However I suspect that many undergraduates may prefer a presentation 
which is less abstract and closer to nineteenth century ideas, reserving
more advanced texts such as Frankel for later study.  
As a graduate text or for a second course, Frankel would surely
be a reasonable choice, though I should say that I'm not enough of an 
expert on differential geometry to judge the suitability
of its choice of topics in this vast field.     

	The book's title, "The Geometry of Physics", suggests its orientation
but might possibly mislead some.  This is a mathematics text, 
not a physics text.  Some applications to physics are presented to illustrate 
the mathematics, but the presentations are usually abbreviated and 
often superficial.  Physics students hoping for a direct and 
seamless connection between Frankel's mathematics and 
presentations of similar material in their physics courses may be disappointed.

	Regarding the superficiality, there is a short introduction to 
general relativity which I did not read because I was already familiar
with the subject.  Therefore, I won't attempt to definitively judge 
how useful it might be to a beginner, but my guess is that 
most beginners would be better off with a more extensive introduction.  
This should not be taken as a criticism of the book's treatment, 
but instead as a caveat that it may not be what all readers might hope for. 
Perhaps  Frankel's 1974 book "Gravitational Curvature" (which I haven't read)
might provide a more relaxed introduction. 


	A summary of the above was already presented in the introductory
paragraph, but for a graceful conclusion I state it once again.
This is a superb book from which I learned much.  Inevitably, it will not
meet the needs or hopes of all readers, and I have tried to indicate some
of its limitations.  But I do think that almost anyone with an appropriate
background can learn much of value from this book.  I doubt that many will
regret adding it to their libraries even if they don't read the whole book.