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) . A Russian translation was published in 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. 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
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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},
}