The new math post, Tensors, is here. It starts like that:
Tensors
Originally, while writing the previous post, I was convinced that this one should be devoted to the Grassmann (or exterior) algebra. I even began drafting it and managed about half a page before realizing that, if the promise of a truly “gentle” approach is to be taken seriously, there is an earlier stop on the journey. Before exterior algebra can comfortably appear on stage, one really ought to say a few clear words about tensors in general.
Moreover, since the plan is to reach Maxwell’s equations at some point, we will also need pseudo-tensors and tensor densities—those slightly more exotic geometric objects that insist on transforming with a twist. So, in this post, tensors and their “relatives” will make their entrance; Grassmann will simply have to wait his turn in the queue of structures. After all, even in mathematics, good manners suggest introducing the family before discussing the exterior.
Then there is an illustration:
Afternotes:
11-12-25 7:30 Laura posted on Substack:
Afternotes
16-12025 9:39 Yesterday O asked Perplexity AI the following question:
Prompt: Please, speculate on possible reasons why there is no English translation of the excellent (in my opinion) book by Sulanke, Wintgen, Differentialgeometrie und Faserbündel. It is strange. So try probable guesses.
Reply: There actually is a translation, but not into English: there is a Russian edition by Mir (“Differential geometry and fibre bundles”, Moscow 1975), explicitly listed as a translation of the 1972 German original. What seems to be missing is precisely an English edition from a major Western publisher. What the record suggests Some bibliographies and course pages loosely refer to an “English translation”, but tracing the citation usually leads back either to the Russian Mir edition or to generic encyclopedia entries where terminology from Sulanke–Wintgen is reused, not to an actual English book. Standard reference lists in differential geometry tend to cite the original Birkhäuser volume rather than any English version, which strongly suggests that if an English text exists at all, it is either unpublished, very limited in circulation, or simply a misdescription of the Mir translation. Market and timing factors When the book appeared (early 1970s), the Anglophone market for differential geometry and bundles was already being served by competing texts (e.g. Bishop–Crittenden, Kobayashi–Nomizu, et al.), so a commercial English translation of another advanced monograph from German would have looked like a modest, risky project to publishers. At the same time, the effective audience for a rigorous, relatively abstract geometry text was small, and many of those readers could already handle German or French, reducing the perceived need for an English edition. Rights, publisher, and author-side issues Translation rights normally require both author (or estate) and publisher alignment, and for a mid-list specialized monograph there may simply never have been a champion—on the author side or at Birkhäuser—to push an English version through. Once the original German publisher and the Soviet Mir edition had recouped their costs, there would be little structural pressure to invest in another translation decades later, especially with changing ownership structures in academic publishing. Shifts in pedagogy and fashion By the time global demand for “standard” bundle-based differential geometry in English really matured (late 1970s–1990s), newer texts had appeared that matched evolving tastes in notation, category-theoretic language, and connections-with-physics presentations, gradually pushing Sulanke–Wintgen into the role of a respected but “classical” reference. Publishers are typically more inclined to commission new books that reflect current pedagogical fashions than to translate older treatments whose style might be viewed as slightly dated, even if mathematically excellent. The “good but not canonical” paradox Finally, there is the familiar paradox that many mathematically elegant books never become “canonical” because they sit just off the main curricular paths, which in turn makes translation projects hard to justify economically. Citations to Sulanke–Wintgen appear regularly in the literature, but almost always as one item among several in reference lists, not as the go‑to textbook, which is exactly the profile that tends to strand a book in its original language. In that sense, the absence of an English edition is less “mysterious” than just a consequence of how conservative and path‑dependent the textbook and translation market is, especially in differential geometry.
I asked Grok the same question. The answer was similar:
While it's indeed puzzling that a well-regarded book like Differentialgeometrie und Faserbündel (1972) by Rolf Sulanke and Peter Wintgen lacks an official English translation—especially given its influence in differential geometry and mentions of translations into other languages like Polish and possibly Russian—here are some probable speculations based on historical, academic, and publishing contexts. These are educated guesses, as no definitive explanations appear in public records.
- Cold War Publishing Barriers: The book was published in East Germany (GDR) by VEB Deutscher Verlag der Wissenschaften, a state-controlled publisher, during the height of the Cold War. Co-publishing with Birkhäuser in Switzerland allowed some Western access, but political divisions likely hindered collaborations with English-language publishers in the US or UK. Intellectual property rights, export restrictions, or ideological scrutiny could have made an English edition logistically challenging or unappealing to Western firms wary of dealing with communist entities.
- Limited Market Demand Due to Niche Appeal and Competition: By the early 1970s, fiber bundles and differential geometry were specialized topics with a relatively small global audience, primarily academics in Europe where German-reading proficiency was common among mathematicians. English alternatives already existed, such as Norman Steenrod's The Topology of Fibre Bundles (1951), Dale Husemöller's Fibre Bundles (1966), or Michael Spivak's multi-volume A Comprehensive Introduction to Differential Geometry (starting in 1969). Publishers might have seen little commercial incentive to translate another text when these covered similar ground more accessibly or comprehensively.
- Authors' Regional Prominence: Sulanke and Wintgen were based in East Germany (Humboldt University in Berlin), and while respected in their field—Wintgen for contributions like the Wintgen inequality—their work didn't achieve the international stardom of figures like Spivak or Kobayashi. Without a high-profile advocate (e.g., a Western mathematician pushing for translation), the book may have remained a "hidden gem" confined to German-speaking circles or those willing to tackle the original.
- High Costs and Complexity of Translation: Mathematical texts require precise, expert translation to preserve notation, proofs, and nuances, which is expensive and time-intensive. In the pre-digital era, without strong sales projections, publishers might have balked at the investment. The book's length (around 300 pages) and its origins partly as a university thesis or lecture notes could have made it seem less "essential" for the effort compared to broader treatises.
- Linguistic and Cultural Factors: German was still a key language for mathematics in the mid-20th century, especially in geometry and topology (think Gauss, Riemann). Many serious students and researchers learned it as a second language, reducing urgency for English versions. By the time English fully dominated STEM publishing in the 1980s–90s, the book was over a decade old and potentially overshadowed by newer works, missing the translation window.
- Rights or Archival Issues: Post-reunification Germany (after 1989) saw many East German publications archived or reissued in original form, but without active interest from heirs, estates, or publishers, no one initiated an English project. If the book drew from collaborative lectures, authorship complexities might have deterred updates or translations.
Interestingly, some course syllabi (e.g., from Ohio State University) claim "an English translation exists," but this appears unsubstantiated—no published edition turns up in major catalogs like WorldCat, Springer, or Amazon. It could refer to unofficial, private translations circulating in academic networks (perhaps by the professor himself), partial adaptations in English notes (like those referencing the book in Andrew Hirshfeld's translated lectures), or even a mix-up with the Polish edition. If true, it might be an obscure, unpublished version, which wouldn't explain the broader absence.Overall, the lack seems more a quirk of timing, geopolitics, and market dynamics than any flaw in the book's quality. If demand surged today (e.g., via open-access initiatives), a translation could still happen, but it's unlikely without a champion.
Then I asked about the alleged Polish translation (I do have Russian edition) that Grok has mentioned, and Grok apologized for lying about it. My own guess that the book was simply "too good", and would create a serious competition for Western authors.
Then I had a further chat with Grok:
Prompt: You wrrote above that Bishop and Go;dgebr in their book discuss weights in tensor transformation rules. I have this book, I am looking at Chapter 2 that you have mentioned, but I am not able to find anything related to tensor densities. Can you help me?
Grok: I apologize for any confusion in my earlier summary—the book uses the term "relative tensors" (a common synonym for tensor densities, especially in older mathematical literature) rather than "tensor densities" explicitly. This is why you might not have spotted it under that exact phrasing in Chapter 2.Upon cross-referencing multiple sources that cite Bishop and Goldberg's work (including differential geometry notes and tensor calculus texts), the discussion of relative tensors (i.e., objects transforming as standard tensors but with an additional factor of the Jacobian determinant raised to a power ω, the weight) appears in Chapter 2, specifically around sections on transformation laws and invariants (likely Sections 2.13–2.14, based on detailed tables of contents and excerpts). Look for subsections dealing with "relative tensors," "tensors of weight ω," or extensions of standard transformation rules that incorporate determinants or Jacobians beyond the usual contravariant/covariant changes. For example:Relative tensors of arbitrary integer weight ω are defined via transformation laws like T′=∣detJ∣ωTT' = |\det J|^\omega TT' = |\det J|^\omega T
, where J is the Jacobian matrix (or its absolute value for even/odd distinctions in some contexts).
The book builds this on the algebraic foundations of multilinear maps and tensor spaces, showing how weights arise in coordinate changes to maintain invariance in certain operations.
If your edition has an index, check entries for "relative tensor," "weight (tensor)," "transformation law (relative)," or "Jacobian in transformations." Invariant operations (e.g., contractions adjusted for weight, or conversions to absolute tensors via metric determinants) are touched on there and expanded in Chapter 3 (Vector Analysis on Manifolds) and Chapter 4 (Integration Theory), where weights ensure coordinate-independent integrals (e.g., via volume forms or oriented densities implicit in exterior algebra and Hodge duality from Sections 2.18–2.22).If you quote a specific page or subsection from Chapter 2 where you're stuck, or describe what terms you see around transformation laws, I can refine this further. Alternatively, if your copy lacks an index or the term isn't there, it might be an implicit treatment through examples like the Levi-Civita symbol or determinants (Section 2.19), which behave as densities of weight 1 or -1.
Prompt: I have made a full text search, and there is no concept of "relative tensor" in this book either.
Grok: I apologize for the inaccuracy in my earlier description—you're absolutely right, and your full-text search confirms it: Bishop and Goldberg's Tensor Analysis on Manifolds (1968/1980) does not discuss tensor densities (or relative tensors) of arbitrary weight, nor the associated invariant operations in the general sense.
The book treats tensors strictly as multilinear maps (function-theoretic in Chapter 1, algebraic in Chapter 2), with standard transformation laws under coordinate changes (no extra Jacobian factors for weights). It covers oriented volume elements and the Levi-Civita symbol implicitly in later chapters (e.g., through alternating tensors and exterior algebra in Sections 2.18–2.22, and integration in Chapter 4), but without introducing weighted densities or general operations on them (like weighted Lie derivatives or arbitrary-weight Hodge duality).The confusion on my part stemmed from associating the book's strong treatment of differential forms and integration on manifolds (where densities naturally arise for invariant measures, e.g., via in pseudo-Riemannian cases) with an explicit discussion of weighted objects. Many texts build densities on top of pure tensor algebra, but Bishop and Goldberg keep it coordinate-free and absolute (true tensors only), handling invariance through forms and orientation rather than weights.For explicit coverage of arbitrary-weight tensor densities and their invariants, better alternatives include:
\sqrt{|g|}- Schouten’s Ricci-Calculus (1954) or Tensor Analysis for Physicists—classic sources for "relative tensors" of weight (w).
- Lee’s Introduction to Smooth Manifolds (brief mentions in tensor chapters).
- More directly in GR contexts: Weinberg’s Gravitation and Cosmology or specialized appendices in texts like Nakahara’s Geometry, Topology and Physics.
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