My first encounter with him was an indirect one. It came about in the summer of 1995 when I was attempting to work through his short and elegant paper on the Carathéodory metric of an annulus in the complex plane. It was a paper that I had come upon purely by accident—a fortuitous one as I now realize. His name had been unfamiliar to me then, but the address for correspondence was not—it read “Tata Institute of Fundamental Research, Colaba, Bombay 400005, India”. The style of writing was direct but demanded a focused engagement. Intrigued both by the beautiful explicit formula for the Carathéodory metric and the unfamiliar author, who was based in a city that was so familiar, I decided to look around for his other papers. With a little bit of effort, there was enough to think about. His work touched upon many foundational topics in several complex variables, the theory of Riemann surfaces, deformation theory and analytic geometry—a diverse spread by any standards. This diversity, as I was to begin learning a decade later, was neither forced nor born of any compulsion. His work was bound by a common thread of eloquent brevity and was vitally alive in his ever cheerful personality—a personality marked with easy humor, warmth and care even towards those he was meeting for the first time.
Our first meeting was a brief one; so brief that I remember very little of what transpired. However, I’m sure of some things—that it was towards the end of 2005, that it was in a corridor of the Department of Mathematics at the Indian Institute of Science and, more importantly, I remember his gentle request of addressing him either as “Raya Simha” or just “Simha”. I chose the latter, perhaps being mindful of the number of years that separated us. He agreed and from that moment onward till we met in early September 2017, a little before his untimely demise, he was “Simha” to me on all platforms of engagement, personal or otherwise. It did not take very long for the years that separated us to tumble down a column of time, never to rise again. Meeting him was always a pleasure and our conversations were never bland. They ranged from the Levi problem, the Weierstrass preparation theorem, the boundaries of Siegel domains in complex dynamics, Sullivan’s proof of Fatou’s No Wandering Domains conjecture, quasiconformal mappings and the Beltrami equation, to name a few, to his favourite detective stories and food. Amidst all this, there were of course jokes and the occasional story relating to his travels in and around the country. Moments of silence were few and far apart but never awkward and quite often would be filled with his trademark humming. Yes, he was fond of music and when I once asked what he was humming, he replied, “Oh, in the past it was just about anything and everything, but mostly Indian classical music now.”
Roddam Raya Simha was born on 3 December 1937 in Bangalore. For his elementary education, he attended Acharya Pathashala, a school that was established in 1935 in the Gandhi Bazaar area and still runs today. Roddam Narasimha, his elder brother who also studied in the same school, recalls that “it valued traditional scholarship and was full of deeply committed teachers”. Simha completed his SSLC exam in 1951 and was ranked tenth in the state merit list. Subsequently, he enrolled in National College, Bangalore and completed the Inter Public Exam in 1953. He obtained his B.Sc. (Honors) (1956) and M.Sc. degrees (1957) in Mathematics from Central College, Bangalore and this was followed by a year-long stint as a Lecturer in BMS College of Engineering, Bangalore. He joined the School of Mathematics at TIFR in July 1958. It was here that his scholarly disposition came to the fore. He studied several complex variables with Raghavan Narasimhan and eventually wrote a thesis under M.S. Narasimhan’s supervision.
Two of his contributions that stand out for their elegance are the explicit formula for the Carathéodory metric of an annulus and his proof of the inequivalence of the ball and polydisc in higher dimensions. It is interesting to note that both contributions address matters that lie at the heart of several complex variables. Apart from these two, which we will discuss in some detail, his other noteworthy and interesting contributions have been listed at the end.
Carathéodory Metric of the Annulus: One version of the classical Schwarz lemma that is encountered in a first course on complex analysis says that every holomorphic map from the unit disc to itself is distance decreasing in the hyperbolic metric. When applied to automorphisms, this property implies that the hyperbolic metric is conformally invariant. For domains whose universal cover is the unit disc, these properties continue to hold since the hyperbolic metric is well defined on it. These properties make the hyperbolic metric an integral ingredient in complex analysis.
There are several ways to create such a metric in higher dimensions. One such is the Carathéodory metric. It is a biholomorphically invariant metric on a complex manifold M that was first defined in 1927. Here is the definition: the Carathéodory distance between a pair of points x, y \in M is the supremum of the hyperbolic distance between f(x) and f(y) where the supremum is taken over all possible holomorphic maps from M to the unit disc in the plane. It may happen that the only global holomorphic functions on M are the constants, in which case the Carathéodory distance is not really a distance after all. But there are many examples of M on which this is a distance and an interesting one at that—for example, let M be a bounded domain in \mathbb C^n. If M is the unit disc in the plane, the Carathéodory metric is the hyperbolic metric.
A dual metric, defined in terms of holomorphic maps from the unit disc into M, was introduced by S. Kobayashi in 1970, whose monograph Hyperbolic Manifolds and Holomorphic Mappings contained the basic properties of this new distance. This is now called the Kobayashi metric and has proved to be very useful in many questions in complex geometry. Again, for the unit disc, the Kobayashi metric is the hyperbolic metric but the Carathéodory and the Kobayashi metrics differ in general. Kobayashi’s monograph also contained a discussion on how these two metrics are related and in it he asks for a description of the Carathéodory metric of an annulus in the plane—here, it should be noted that the Kobayashi metric for the annulus was known to him and is contained in his monograph.
In [4], Simha observed that for an annulus, the extremal function that realizes the Carathéodory distance between a pair of points is exactly the Ahlfors map. Recall that for a domain with more than one but finitely many boundary components, the Ahlfors map exhibits the domain as a ramified covering of the unit disc of order equal to the number of boundary components. In fact, each boundary component is mapped to the unit circle. Thus, this map may be regarded as an analogue of the Riemann map that uniformizes simply connected domains by the unit disc.
In particular, the Ahlfors map for the annulus is a ramified covering of the unit disc of order two and unique up to a multiplicative constant of modulus one. For an annulus, it is not too difficult to write down this function in terms of the classical theta functions. This gives an explicit formula for the Carathéodory metric of an annulus, whereas nothing like this is known for planar domains with higher connectivity.
The Inequivalence of the Ball and Polydisc in Higher Dimensions: The Riemann mapping theorem fails in higher dimensions. This follows, for instance, from the fact that a ball and polydisc are holomorphically inequivalent in dimensions bigger than one. Thus the purely topological property of being simply connected does not imply holomorphic equivalence in higher dimensions.
There are many proofs of this theorem which is usually attributed to Poincaré. Though Poincaré wrote an influential paper on the holomorphic equivalence problem (the problem of deciding when a given pair of domains are holomorphically equivalent), there is no explicit mention of this claim in it. However, he did compute the holomorphic automorphism group of the ball in \mathbb C^2 and showed that it depends on 8 real parameters. Computing the automorphism group of the bidisc is relatively easier; it consists of Möbius transformations in each variable composed with a possible interchange of variables. In all, this depends on 6 real parameters and hence the two domains cannot be biholomorphic. The proof extends to higher dimensions as well. Another way to see this is to compute the Carathéodory (or the Kobayashi) metric on these two domains using the Schwarz lemma. On the ball, these metrics are real analytic while on the polydisc they are merely continuous near a large set of points. The holomorphic inequivalence of the ball and polydisc follows.
In [8], Simha gave a completely elementary proof of this fact using only the classical Schwarz lemma. The main step is to show that every holomorphic map from the ball to the polydisc (or from the polydisc to the ball) that preserves the origin, has the property that its differential at the origin, as a linear map, maps the ball into the polydisc (or the polydisc to the ball). This is done by applying the classical Schwarz lemma to the restriction of this map to one-dimensional slices passing through the origin. Once this is established, any biholomorphism between the ball and the polydisc (since the polydisc is evidently homogeneous, it is always possible to arrange that the origin is a fixed point) would have the property that its differential at the origin is a linear equivalence between them. But this is evidently not possible since the boundary of the polydisc contains open pieces of complex lines while that of the ball does not.
The Lecture Note Series and Studies in Mathematics are two well-known book series of the School of Mathematics at TIFR. These are based on the lectures delivered by visiting mathematicians (many a time for extended periods), with notes being written by younger students. Simha contributed in writing up the lectures of Michel Hervé (Several Complex Variables: Local Theory) and J. Koszul (Lectures on Groups of Transformations). He was also part of the team that wrote the TIFR Mathematical Pamphlet on Riemann Surfaces.
He was blessed with an unusual flair for languages
Blessed with an unusual flair for languages, he found the time and energy to translate two texts on two very different topics, written in very different styles in two different languages. One was translated from the Russian (Differential Forms Orthogonal to Holomorphic Functions or Forms and Their Properties by L.A. Aizenberg and Sh.A. Dautov) while the other from German (Compact Riemann Surfaces: An Introduction to Contemporary Mathematics by Jürgen Jost). Then there is [3], a paper written by him in German. Keeping aside the skills required in effectively translating a mathematical text or for that matter writing a paper in German, he was also knowledgeable about the linguistic nuances of that language. MathSciNet, a database of the American Mathematical Society (AMS), bears testimony to his efforts in reviewing and translating several papers from Russian. In addition, he also had a working knowledge of French. Closer home, apart from his mother tongue Kannada, he adored Bengali, was fluent in Telugu and comfortable in Hindi. He read and enjoyed Urdu poetry and found space for managing Marathi and Tamil too.
Above all, Simha was a committed teacher and had a wonderful and engaging style of lecturing. He spoke in a slow measured tone without strain or abruptness and had the capacity and sensitivity to know whether the audience was with him or not. He wrote in beautiful cursive style both on the black board and on paper. I have very pleasant and warm memories of spending time with him during several Annual Foundation School events held at Delhi University, Bhaskaracharya Pratisthana in Pune and the University of Hyderabad. Students found him very approachable and I’m sure they learnt many things from his lectures.
Simha was a committed teacher and had a wonderful and engaging style of lecturing
Particularly vivid are my memories of two such schools held at Delhi University two years in a row in 2009 and 2010. Our stay overlapped for about a week on both occasions. After the day’s work was done, a few of us would take the Yellow Line from Vishwavidyala Metro Station to Rajiv Chowk. The conversation would invariably start with something in complex analysis and as the mood of the evening changed, move effortlessly to the detective stories of Father Brown (I was completely clueless here) and then to the merits of palak pooris over plain ones and the chaats of Chandni Chowk.
Simha passed away on 14 September 2017. A memorial gathering was held on 24 September in Padmanabhanagar, Bangalore. Despite starting from home quite early, I reached about twenty minutes late. As I was crossing the road to reach the event hall, a part of me wanted to resume the conversation on the Levi problem that we had had just a couple of weeks ago. But my heart knew that this would not be one of our regular meetings. Sitting on a table, with two lighted lamps on either side, was a beautiful photograph of Simha. He was smiling and though not physically present, he was everywhere. After the musical recitals were over, several members of his family reminisced about him. What emerged was a portrait of a person full of warmth and care—a person sensitive to the needs of those around him, be it family or otherwise, and one who gave of himself freely. M.S. Narasimhan recalled his association with him and said that theirs had not been an ordinary superviser–student relation, for Simha had been mathematically mature even during his early student days at TIFR. He added that Simha’s in-depth knowledge of various areas in mathematics had benefited many people at TIFR.
A few days later, sitting in my office, as my thoughts were drifting, I found myself thinking about him. I could not help but recall what Gibran wrote in his masterpiece, The Prophet: “And there are those who give and know not pain in giving, nor do they seek joy, nor give with mindfulness of virtue; They give as in yonder valley, the myrtle breathes its fragrance into space.” Simha was certainly one of “those”. He was, after all, a rare breed.
References
- [1] R.R. Simha. On the Complement of a Curve on a Stein Space of Dimension Two. Math. Z. 1963. 82: 63–66
- [2] M.S. Narasimhan, R.R. Simha. Manifolds with Ample Canonical Class. Invent. Math. 1968. 5: 120–128
- [3] R.R. Simha. Über die kritischen Werte gewisser holomorpher Abbildungen. Manuscripta Math. 1970. 3: 97–104
- [4] R.R. Simha. The Carathéodory Metric of the Annulus. Proc. Amer. Math. Soc. 1975. 50: 162–166
- [5] B.V. Limaye, R.R. Simha. Deficiencies of Certain Real Uniform Algebras. Canad. J. Math. 1975. 27: 121–132
- [6] R.R. Simha. On the Analyticity of Certain Singularity Sets. J. Indian Math. Soc. (N.S.) 39 (1975), 281–283 (1976)
- [7] R.R. Simha. Certain Cones are Not Set-Theoretic Complete Intersections. Arch. Math. (Basel) 1976. 27(2): 169–171
- [8] R.R. Simha. Holomorphic Mappings between Balls and Polydiscs. Proc. Amer. Math. Soc. 1976. 54: 241–242
- [9] R.R. Simha. Algebraic Varieties Biholomorphic to \mathbb C^{\ast} \times \mathbb C^{\ast}. Tôhoku Math. J. (2) 1978. 30(3): 455–461
- [10] J.R. Choksi, R.R. Simha. Measurable Transformations on Homogeneous Spaces. Studies in Probability and Ergodic Theory, pp. 269–286, Adv. in Math. Suppl. Stud., 2, Academic Press, New York-London, 1978
- [11] R.R. Simha. Holomorphic Maps into Compact Complex Spaces. Arch. Math. (Basel) 1982. 39(3): 262–263
- [12] R.R. Simha. Biholomorphic Maps between Circular Domains in Complex Banach Spaces. J. Math. Phys. Sci. 1987. 21(3): 261–269
- [13] R.R. Simha. The Monodromy Representations of Projective Structures. Arch. Math. (Basel) 1989. 52(4): 413–416
- [14] R.V. Gurjar, R.R. Simha. Some Results on the Topology of Varieties Dominated by \mathbb C^n. Math. Z. 1992. 211(2): 333–340
- [15] R.R. Simha, V. Srinivas. Riemann Surfaces. Analysis, Geometry and Probability. pp. 198–273, Texts Read. Math., 10, Hindustan Book Agency, Delhi, 1996.