M. Atiyah, R. Bott, and V. K. Patodi :
“On the heat equation and the index theorem ,”
Matematika, Moskva
17 : 6
(1973 ),
pp. 3–48 .
Russian translation of an article in Invent. Math. 19 :4 (1973)
Zbl
0364.58016 article
People
BibTeX
@article {key0364.58016z,
AUTHOR = {Atiyah, Michael and Bott, Raoul and
Patodi, V. K.},
TITLE = {On the heat equation and the index theorem},
JOURNAL = {Matematika, Moskva},
VOLUME = {17},
NUMBER = {6},
YEAR = {1973},
PAGES = {3--48},
NOTE = {Russian translation of an article in
\textit{Invent. Math.} \textbf{19}:4
(1973). Zbl:0364.58016.},
}
M. Atiyah, R. Bott, and V. K. Patodi :
“On the heat equation and the index theorem Invent. Math.
19 : 4
(1973 ),
pp. 279–330 .
Dedicated to Sir William Hodge on his 70th birthday.
Errata were published in Invent. Math. 28 :3 (1975)Matematika 17 :6 (1973)
MR
0650828 Zbl
0257.58008 article
Abstract
People
BibTeX
@article {key0650828m,
AUTHOR = {Atiyah, M. and Bott, R. and Patodi,
V. K.},
TITLE = {On the heat equation and the index theorem},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {19},
NUMBER = {4},
YEAR = {1973},
PAGES = {279--330},
DOI = {10.1007/BF01425417},
NOTE = {Dedicated to Sir William Hodge on his
70th birthday. Errata were published
in \textit{Invent. Math.} \textbf{28}:3
(1975). A Russian translation was published
in \textit{Matematika} \textbf{17}:6
(1973). MR:0650828. Zbl:0257.58008.},
ISSN = {0020-9910},
}
M. F. Atiyah, V. K. Patodi, and I. M. Singer :
“Spectral asymmetry and Riemannian geometry Bull. London Math. Soc.
5
(July 1973 ),
pp. 229–234 .
MR
331443 Zbl
0268.58010 article
Abstract
People
BibTeX
If \( A \) is a positive self-adjoint elliptic (linear) differential operator on a compact manifold then it has a discrete spectrum consisting of positive eigenvaues \( \{\lambda\} \) . In analogy with the classical Riemann zeta-function one can define, for \( \operatorname{Re}(s) \) large,
\[ \zeta_A(s) = \operatorname{Tr} A^{-s} = \sum\lambda^{-s}. \]
@article {key331443m,
AUTHOR = {Atiyah, M. F. and Patodi, V. K. and
Singer, I. M.},
TITLE = {Spectral asymmetry and {R}iemannian
geometry},
JOURNAL = {Bull. London Math. Soc.},
FJOURNAL = {The Bulletin of the London Mathematical
Society},
VOLUME = {5},
MONTH = {July},
YEAR = {1973},
PAGES = {229--234},
DOI = {10.1112/blms/5.2.229},
NOTE = {MR:331443. Zbl:0268.58010.},
ISSN = {0024-6093},
}
M. F. Atiyah :
“The heat equation in Riemannian geometry (after Patodi, Gilkey, etc.) ,”
pp. 1–11
in
Séminaire Bourbaki 1973/1974 .
Lecture Notes in Mathematics 431 .
Springer (Berlin ),
1975 .
Exposé no. 436.
MR
0431286 incollection
People
BibTeX
@incollection {key0431286m,
AUTHOR = {Atiyah, M. F.},
TITLE = {The heat equation in {R}iemannian geometry
(after {P}atodi, {G}ilkey, etc.)},
BOOKTITLE = {S\'eminaire {B}ourbaki 1973/1974},
SERIES = {Lecture Notes in Mathematics},
NUMBER = {431},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1975},
PAGES = {1--11},
NOTE = {Expos\'e no.~436. MR:0431286.},
ISSN = {0075-8434},
ISBN = {9780387070230},
}
M. Atiyah, R. Bott, and V. K. Patodi :
“Errata to: ‘On the heat equation and the index theorem’ Invent. Math.
28 : 3
(1975 ),
pp. 277–280 .
Errata for article in Invent. Math. 19 :4 (1973)
MR
0650829 Zbl
0301.58018 article
Abstract
People
BibTeX
The joint paper of the above title which appeared in Inventiones Math. 19 , 279–330 (1973), though correct in principle, contained some technical errors which we shall here explain and rectify. Our thanks are due to D. Epstein, Y. Colin de Verdiére and A. Vasquez whose computations and queries alerted us to our errors.
@article {key0650829m,
AUTHOR = {Atiyah, M. and Bott, R. and Patodi,
V. K.},
TITLE = {Errata to: ``{O}n the heat equation
and the index theorem''},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {28},
NUMBER = {3},
YEAR = {1975},
PAGES = {277--280},
DOI = {10.1007/BF01425562},
NOTE = {Errata for article in \textit{Invent.
Math.} \textbf{19}:4 (1973). MR:0650829.
Zbl:0301.58018.},
ISSN = {0020-9910},
}
M. F. Atiyah, V. K. Patodi, and I. M. Singer :
“Spectral asymmetry and Riemannian geometry, I Math. Proc. Cambridge Philos. Soc.
77
(1975 ),
pp. 43–69 .
MR
397797 Zbl
0297.58008 article
Abstract
People
BibTeX
The main purpose of this paper is to present a generalization of Hirzebruch’s signature theorem for the case of manifolds with boundary. Our result is in the framework of Riemannian geometry and can be viewed as analogous to the Gauss–Bonnet theorem for manifolds with boundary, although there is a very significant difference between the two cases which is, in a sense, the central topic of the paper.
@article {key397797m,
AUTHOR = {Atiyah, M. F. and Patodi, V. K. and
Singer, I. M.},
TITLE = {Spectral asymmetry and {R}iemannian
geometry, {I}},
JOURNAL = {Math. Proc. Cambridge Philos. Soc.},
FJOURNAL = {Mathematical Proceedings of the Cambridge
Philosophical Society},
VOLUME = {77},
YEAR = {1975},
PAGES = {43--69},
DOI = {10.1017/S0305004100049410},
NOTE = {MR:397797. Zbl:0297.58008.},
ISSN = {0305-0041},
}
M. F. Atiyah, V. K. Patodi, and I. M. Singer :
“Spectral asymmetry and Riemannian geometry, II Math. Proc. Cambridge Philos. Soc.
78 : 3
(November 1975 ),
pp. 405–432 .
MR
397798 Zbl
0314.58016 article
Abstract
People
BibTeX
@article {key397798m,
AUTHOR = {Atiyah, M. F. and Patodi, V. K. and
Singer, I. M.},
TITLE = {Spectral asymmetry and {R}iemannian
geometry, {II}},
JOURNAL = {Math. Proc. Cambridge Philos. Soc.},
FJOURNAL = {Mathematical Proceedings of the Cambridge
Philosophical Society},
VOLUME = {78},
NUMBER = {3},
MONTH = {November},
YEAR = {1975},
PAGES = {405--432},
DOI = {10.1017/S0305004100051872},
NOTE = {MR:397798. Zbl:0314.58016.},
ISSN = {0305-0041},
}
M. F. Atiyah, V. K. Patodi, and I. M. Singer :
“Spectral asymmetry and Riemannian geometry, III Math. Proc. Cambridge Philos. Soc.
79 : 1
(1976 ),
pp. 71–99 .
MR
397799 Zbl
0325.58015 article
Abstract
People
BibTeX
In Parts I and II of this paper [1975a; 1975b] we studied the ‘spectral asymmetry’ of certain elliptic self-adjoint operators arising in Riemannian geometry. More precisely, for any elliptic self-adjoint operator \( A \) on a compact manifold we defined
\[ \eta_A(s) = \sum_{\lambda\neq 0}\operatorname{sign} \lambda \,|\lambda|^{-s}, \]
where \( \lambda \) runs over the eigenvalues of \( A \) . For the particular operators of interest in Riemannian geometry we showed that \( \eta_A(s) \) had an analytic continuation to the whole complex \( s \) -plane, with simple poles, and that \( s = 0 \) was not a pole. The real number \( \eta_A(0) \) , which is a measure of ‘spectral asymmetry’, was studied in detail particularly in relation to representations of the fundamental group.
@article {key397799m,
AUTHOR = {Atiyah, M. F. and Patodi, V. K. and
Singer, I. M.},
TITLE = {Spectral asymmetry and {R}iemannian
geometry, {III}},
JOURNAL = {Math. Proc. Cambridge Philos. Soc.},
FJOURNAL = {Mathematical Proceedings of the Cambridge
Philosophical Society},
VOLUME = {79},
NUMBER = {1},
YEAR = {1976},
PAGES = {71--99},
DOI = {10.1017/S0305004100052105},
NOTE = {MR:397799. Zbl:0325.58015.},
ISSN = {0305-0041},
}
M. F. Atiyah and R. Bott :
“Yang–Mills and bundles over algebraic curves ,”
pp. 11–20
in
Geometry and analysis: Papers dedicated to the memory of V. K. Patodi .
Indian Academy of Sciences (Bangalore ),
1980 .
Republished in Proc. Indian Acad. Sci. Math. Sci. 90 :1 (1981)
MR
592249 Zbl
0482.14007 incollection
People
BibTeX
@incollection {key592249m,
AUTHOR = {Atiyah, M. F. and Bott, R.},
TITLE = {Yang--{M}ills and bundles over algebraic
curves},
BOOKTITLE = {Geometry and analysis: {P}apers dedicated
to the memory of {V}.~{K}. {P}atodi},
PUBLISHER = {Indian Academy of Sciences},
ADDRESS = {Bangalore},
YEAR = {1980},
PAGES = {11--20},
NOTE = {Republished in \textit{Proc. Indian
Acad. Sci. Math. Sci.} \textbf{90}:1
(1981). MR:592249. Zbl:0482.14007.},
ISBN = {9780387102702},
}
M. F. Atiyah and R. Bott :
“Yang–Mills and bundles over algebraic curves 11–20
in
Geometry and analysis: Papers dedicated to the memory of V. K. Patodi ,
published as Proc. Indian Acad. Sci., Math. Sci.
90 : 1 .
Indian Academy of Sciences (Bangalore ),
1981 .
MR
653942 Zbl
0499.14005 incollection
People
BibTeX
@article {key653942m,
AUTHOR = {Atiyah, M. F. and Bott, R.},
TITLE = {Yang--{M}ills and bundles over algebraic
curves},
JOURNAL = {Proc. Indian Acad. Sci., Math. Sci.},
FJOURNAL = {Proceedings of the Indian Academy of
Sciences. Mathematical Sciences},
VOLUME = {90},
NUMBER = {1},
YEAR = {1981},
PAGES = {11--20},
DOI = {10.1007/BF02867013},
NOTE = {\textit{Geometry and analysis: {P}apers
dedicated to the memory of {V}.~{K}.
{P}atodi}. MR:653942. Zbl:0499.14005.},
ISSN = {0253-4142},
ISBN = {9780387102702},
}
