Handbook of Differential Geometry 1st Edition Dillen
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- 5. Handbook of Differential Geometry 1st Edition Dillen Digital Instant Download Author(s): Dillen, Franki J.E.; Verstraelen, Leopold C.A. ISBN(s): 9780444520524, 044452052X Edition: 1 File Details: PDF, 3.78 MB Year: 2006 Language: english
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- 9. Handbook of Differential Geometry • VOLUME II Editors Franki J.E. Dillen Leopold C.A. Verstraelen Katholieke Universiteit Leuven Department of Mathematics Leuven, Belgium Amsterdam • Boston • Heidelberg • London • New York • Oxford Paris • San Diego • San Francisco • Singapore • Sydney • Tokyo North-Holland is an imprint of Elsevier
- 10. ELSEVIER B.V. Radarweg 29 P.O. Box 211, 1000 AE Amsterdam The Netherlands ELSEVIER Inc. 525 B Street, Suite 1900 San Diego, CA 92101-4495 USA ELSEVIER Ltd The Boulevard, Langford Lane Kidlington, Oxford OX5 1GB UK ELSEVIER Ltd 84 Theobalds Road London WC1X 8RR UK © 2006 Elsevier B.V. All rights reserved. This work is protected under copyright by Elsevier B.V., and the following terms and conditions apply to its use: Photocopying Single photocopies of single chapters may be made for personal use as allowed by national copyright laws. Permission of the Publisher and payment of a fee is required for all other photocopying, including multiple or systematic copying, copying for advertising or promotional purposes, resale, and all forms of document delivery. Special rates are available for educational institutions that wish to make photocopies for non-profit educational classroom use. Permissions may be sought directly from Elsevier’s Rights Department in Oxford, UK: phone (+44) 1865 843830, fax (+44) 1865 853333, e-mail: permissions@elsevier.com. Requests may also be completed on-line via the Elsevier homepage (http://www.elsevier.com/locate/permissions). In the USA, users may clear permissions and make payments through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA; phone: (+1) (978) 7508400, fax: (+1) (978) 7504744, and in the UK through the Copyright Licensing Agency Rapid Clearance Service (CLARCS), 90 Tottenham Court Road, London W1P 0LP, UK; phone: (+44) 20 7631 5555; fax: (+44) 20 7631 5500. Other countries may have a local reprographic rights agency for payments. Derivative Works Tables of contents may be reproduced for internal circulation, but permission of the Publisher is required for external resale or distribution of such material. Permission of the Publisher is required for all other derivative works, including compilations and translations. Electronic Storage or Usage Permission of the Publisher is required to store or use electronically any material contained in this work, including any chapter or part of a chapter. Except as outlined above, no part of this work may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior written permission of the Publisher. Address permissions requests to: Elsevier’s Rights Department, at the fax and e-mail addresses noted above. Notice No responsibility is assumed by the Publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made. First edition 2006 Library of Congress Cataloging in Publication Data A catalog record is available from the Library of Congress. British Library Cataloguing in Publication Data A catalogue record is available from the British Library. ISBN-13: 978-0-444-52052-4 ISBN-10: 0-444-52052-X ∞ The paper used in this publication meets the requirements of ANSI/NISO Z39.48-1992 (Permanence of Paper). Printed in The Netherlands.
- 11. Dedication In memory of S.S. Chern and T. Willmore
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- 13. Preface “Our goal with the volumes which together will constitute the “Handbook of Differen- tial Geometry” is to give a rather complete survey of the field of differential geometry.” Thus reads the opening sentence of the “Handbook of Differential Geometry, Volume I”, and only the presence of the word “rather” saves this goal from being an obvious mis- sion impossible. Let us recall the contents of this Volume I: Differential geometry of webs (M.A. Akivis and V.V. Goldberg), Spaces of metrics and curvature functionals (D.E. Blair), Riemannian submanifolds (B.-Y. Chen), Einstein metrics in dimension four (A. Derdzin- ski), The Atiyah–Singer index theorem (P.B. Gilkey), Survey of isospectral manifolds (C.S. Gordon), Submanifolds with parallel fundamental form (Ü. Lumiste), Sphere the- orems (K. Shiohama), Affine differential geometry (U. Simon), A survey on isoparametric hypersurfaces and their generalizations (G. Thorbergsson), Curves (T. Willmore); with introduction by S.S. Chern. As in Volume I, we allowed the authors in this Volume II as much freedom as possible concerning style and contents. We are confident that the reader will appreciate this prag- matic point of view. Some contributions will emphasize the basics; some will emphasize the classical results; others the recent developments. Needless to say all authors have spent a lot of time and energy in describing their topic, which we appreciate enormously. The contributions to this Volume II are: Some problems on Finsler geometry (J.C. Ál- varez Paiva), Foliations (R. Barre and A. El Kacimi), Symplectic geometry (A. Can- nas da Silva), Metric Riemannian geometry (K. Fukaya), Contact geometry (H. Geiges), Complex differential geometry (I. Mihai), Compendium on the geometry of Lagrange spaces (R. Miron), Certain actual topics on modern Lorentzian geometry (F.J. Palomo and A. Romero). Obviously the whole field of differential geometry is not yet covered in the two volumes of this “Handbook of Differential Geometry”. Some of the authors explicitly mention top- ics that should have been covered, but are not for practical reasons; but also other topics are not (yet) treated sufficiently or not treated at all. Recently Professors Chern and Willmore passed away. Both had a great impact on the development of contemporary geometry and were genuine sources of inspiration, guidance and support for many generations of mathematicians through their books and articles, their fantastic lectures and their warm and truly concerned personal contacts. Together with all authors we gratefully dedicate this book to the memories of Professor S.S. Chern and Professor T.J. Willmore. Franki Dillen and Leopold Verstraelen vii
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- 15. List of Contributors Álvarez Paiva, J.C., Polytechnic University, Brooklyn, NY (Ch. 1). Barre, R., Université de Valenciennes, Valenciennes (Ch. 2). Cannas da Silva, A., Instituto Superior Técnico, Lisboa (Ch. 3). El Kacimi Alaoui, A., Université de Valenciennes, Valenciennes (Ch. 2). Fukaya, K., Kyoto University, Kyoto (Ch. 4). Geiges, H., Universität zu Köln, Köln (Ch. 5). Mihai, I., University of Bucharest, Bucharest (Ch. 6). Miron, R., “Al.I. Cuza” University Iasi, Iasi (Ch. 7). Palomo, F.J., Universidad de Málaga, Málaga (Ch. 8). Romero, A., Universidad de Granada, Granada (Ch. 8). ix
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- 17. Contents Preface vii List of Contributors ix Contents of Volume I xiii 1. Some problems on Finsler geometry 1 J.C. Álvarez Paiva 2. Foliations 35 R. Barre and A. El Kacimi Alaoui 3. Symplectic geometry 79 A. Cannas da Silva 4. Metric Riemannian geometry 189 K. Fukaya 5. Contact geometry 315 H. Geiges 6. Complex differential geometry 383 I. Mihai 7. Compendium on the geometry of Lagrange spaces 437 R. Miron 8. Certain actual topics on modern Lorentzian geometry 513 F.J. Palomo and A. Romero Author Index 547 Subject Index 555 xi
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- 19. Contents of Volume I Preface v Introduction vii List of Contributors ix 1. Differential geometry of webs 1 M.A. Akivis and V.V. Goldberg 2. Spaces of metrics and curvature functionals 153 D.E. Blair 3. Riemannian submanifolds 187 B.-Y. Chen 4. Einstein metrics in dimension four 419 A. Derdzinski 5. The Atiyah–Singer index theorem 709 P.B. Gilkey 6. Survey of isospectral manifolds 747 C.S. Gordon 7. Submanifolds with parallel fundamental form 779 Ü. Lumiste 8. Sphere theorems 865 K. Shiohama 9. Affine differential geometry 905 U. Simon 10. A survey on isoparametric hypersurfaces and their generalizations 963 G. Thorbergsson 11. Curves 997 T. Willmore Author Index 1025 Subject Index 1037 xiii
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- 22. 101 “It’s too much, sir,” he gasped. “No man can go into that stuff and live.” Before Captain West could reply, Sandy Steele had raced down the passageway from the mess hall. “Let me have that raincoat,” he said to the astounded man. “I think I know a way to get that table out.” Still choking, the man took off his coat. Captain West opened his mouth to protest, but then, seeing that Sandy was dead serious, he closed it again and let the determined youth take over. “Jerry!” Sandy called to his chum. “Quick! You get one on, too. Then, you protect me with the fire extinguisher while I swing the ax.” Jerry James nodded. Like his friend, he garbed himself in one of the heavy black slickers, covered his nose and mouth with a soaked cloth, and preceded him into the smoke. Jerry held his extinguisher like a soldier wielding a light machine gun, spraying the flames with a constant stream of thick, white chemicals. Behind him moved Sandy Steele, grasping his ax. The combination that worked so well on the playing fields of their home state of California was now going into action far, far from home, and in a far more serious cause. But it was working just as well! Choking, sputtering, staggering, all but blinded, Sandy Steele charged to the reddish blur he could see a few feet ahead of him in the smoke. Waves of heat rolled against his body and he felt himself going weak. But he lowered his head and struck on.
- 23. 102 Once, a tongue of flame seemed about to gather in volume and leap toward him from the roaring chopping- block. Just in time, a jet of thick white liquid streamed out toward it and smothered it before it could get started. Good old Jerry, Sandy thought. At last, he had made it to within a few feet of the burning table! It was as close as he dared go. Without hesitation, Sandy Steele raised his ax and brought it down, hard. Crash! The table seemed to sway. Sandy raised his arms again, wondering if he would have the strength for another blow. He was thoroughly sick, now—nauseated by that sickening, grease-laden smoke. The effort of his first mighty stroke had all but sapped his strength. Yet, he could not falter now! He had to do it! One more stroke would slice through the remaining wood. Calling upon all his reserves, Sandy Steele rocked backward on his heels, rose on his toes and brought the ax down upon the wood. It was a blow that rang out even above the roar of the flames! Even the weary men gathered in the passageway could hear it. And it severed the table from the thick bolt that had held it to the bulkhead. Sandy Steele jumped back just in time.
- 24. 103 104 With a loud crash and a flashing of sparks and a shooting of flames, the table fell toward him. The momentum of Sandy’s jump sent him staggering backward, off balance. That was how he emerged from the cloud of smoke that separated the excited, yelling crewmen from the fire inside the galley. Behind Sandy, running low and gasping, but still clutching his fire extinguisher, came Jerry James. If someone had not caught Sandy, he would have gone sprawling. As it was, he was having difficulty keeping his legs under him. They seemed to have gone all rubbery from his ordeal. But he clenched his teeth and stayed erect, watching as the crewmen began to drag the blazing table from the galley into the direct play of massed hoses and extinguishers. It sizzled and smoked and sent off clouds of steam as though it were a small volcano, but the fire was at last put out. Then, one by one, all of the other burning articles within the galley were separated from the main body of the fire and doused. The hoses sent streams of lake water splashing against the now-smoldering and smoking bulkheads. The bucket brigade was disbanded, for it was no longer needed. And then, as Sandy Steele felt the youthful vigor of his body swiftly returning, his eyes fell on an object that he dearly wished to preserve for the eyes of Captain West. It was the rum bottle. It lay beside the stove, almost at the exact point where it had fallen from the hand of Mr. Briggs.
- 25. 105 Here was not only the cause of the fire. Here was proof of who really had started it! Sandy slipped from the support of the friendly arms that had grasped him. He bent to pick up an asbestos glove dropped by one of the crewmen. He slipped it on his right hand and walked quickly forward to retrieve the bottle. As he leaned over, he felt himself jostled aside. He nearly fell down again. A tall man stepped in front of him and swung the flat of an ax down on the bottle. He did it deliberately. He shattered the bottle into a hundred pieces. “Why did you do that?” Sandy cried, unable to hide his anger. The man in front of him turned with a wicked smile, and said, “You could have burned yourself on that, Little Lord Show-off—and you’re in enough hot water already.” It was Mr. Briggs.
- 26. 106 CHAPTER NINE Charged with Arson No one was less surprised than Sandy Steele when the order came for him and Jerry James to report to Captain West in his cabin. It was by then close to midnight. Once the fire had been put out, there had remained the task of clearing away the debris and cleaning up. This had occupied the crew for a few more hours, and Sandy and Jerry had not been happy to hear the grumbles about burned suppers and lost sleep or to see the glances of hostility that were directed their way. Mr. Briggs, it seemed, had been as expert in spreading his falsehoods among the crew as he had been in taking them to Captain West. Only Sam had remained friendly, and it had been Sam who had brought the order. “Captain says you two are to report to him right away,” Sam said. He shook his head sadly. “Too bad, boys,” he went on. “If I can read storm signals right, I’d say you were in for it.” “In for it!” Jerry burst out hotly. “Is that what we get for putting out the fire?”
- 27. 107 “Hold it, Jerry,” Sandy said gently, calming his friend down. “That won’t do any good.” He looked at Sam. “I suppose Mr. Briggs is with him?” Sam seemed surprised. “Now, how do you know that?” Sandy’s answer was a grim tightening of his lips. On the subject of Mr. Briggs, he did not trust himself to speak. Sandy wondered how much longer he was going to be able to control his temper. It seemed to him that every time either he or Jerry did something they were supposed to do, even something they really needn’t have done, their only reward was some penalty or a leer from Mr. Briggs or an insult from the skipper. What had begun as a high school boy’s dream of a splendid way to spend the summer seemed to be turning into a nightmare. Sandy let out his breath in a deep sigh. He looked at Jerry and was startled to see the sulky expression on his friend’s normally cheerful countenance. “I’m not going,” Jerry said sullenly. “Wha-a-at?” Sam said, as though he couldn’t believe his ears. “What did you say, young fellow?” Before Jerry could reply, Sandy had propelled him up the passageway and out of earshot. He didn’t want their friend Sam to get the notion that they were mutinous. “Jerry,” he whispered fiercely, “you’ve got to stop talking like that!” “I don’t care!” Jerry said stoutly. “We’ve been pushed around long enough, and now I’ve got to get it off my chest. Listen, Sandy—you know very well what’s going to happen when we get in there with the captain. He’s
- 28. 108 going to accuse us with a lot of lies that he’s heard from the mate. He’ll not only forget that we risked our lives to get at that table, but he’ll turn around and say we started the fire.” “Shhh!” Sandy said, looking around anxiously. Jerry lowered his voice, but he didn’t stop talking. “It’s true! Why, look what he said to you after you rescued poor old Cookie from drowning! He acted as though you’d jumped in just to make him late for the Soo Locks. Honestly, Sandy, I don’t know why you bother—” “Because we’ve got to!” Sandy insisted, squeezing Jerry’s arm. “Don’t you realize that a captain aboard ship is a lot different from a teacher or a football coach? He’s got you in his power, Jerry. His word is law! Really. You can’t disobey him!” “Oh, no?” Jerry said. “If you do,” Sandy warned, “you’ll wind up in jail. I mean it, Jerry. Now is just the time when we’ve got to keep our heads.” He dropped his voice to a whisper. Then he went on: “Captain West must know by now that we’ve found out about him. You remember that Mr. Briggs was out in the passageway, eavesdropping, while we were talking about it. He’s certainly told the skipper. Now, with the fire, he’s got an excuse to do something that will keep us from warning Mr. Kennedy.” Jerry’s eyes widened. “Such as what?” he asked. “Such as locking us up somewhere.” There was a momentary silence, and then Jerry James groaned and said, “Boy, oh, boy, we really are in trouble, aren’t we?”
- 29. 109 Sandy smiled in relief. He could tell by the tone of his friend’s voice that he had gotten over his resentment. With a reassuring squeeze of Jerry’s arm, Sandy continued, “We are. That’s why we’ve got to stay calm. So, whatever you do, Jerry, don’t say or do anything foolish when we get in there with Captain West.” Jerry James’s jaw tightened and he clapped his friend on the arm. “Right,” he said, and then the two of them walked up the passageway and knocked on the door of Captain West’s cabin. “Come in,” the skipper growled. They entered. “What took you so long?” Captain West snapped. “We were delayed,” Sandy said. “Oh,” the skipper mocked, glancing over at his mate, who sat on the bunk. “Did you hear that, Briggs? They were delayed, he says. Well,” he sneered, his voice turning ugly, “you’ll have plenty of time for delays where I’m putting you.” The skipper peered at them with eager expectation, as though he hoped his remarks would goad them into losing their tempers. Observing this, Sandy was inwardly pleased. He realized that the skipper could not be too confident of himself, that he was not sure of how much the youths actually knew—no matter what Mr. Briggs had said to him. “Well?” the skipper roared, crashing his fist down on his desk. “What have you to say to that?”
- 30. 110 111 “Nothing, sir,” Sandy replied evenly. A red flush began to spread over Captain West’s face. But it was supplanted by a cunning look. “Playing doggo, eh?” he muttered. “Well, we’ll see.” He looked over at his mate with a grin, and said, “Now, you just tell that story of yours again, Mr. Briggs.” The mate nodded. “It was this way, sir,” he started, gazing up at the overhead with an expression of shocked innocence. “Just before suppertime, I happened to be passing the galley and saw these two.” He lowered his eyes and jabbed a dirty thumb in the direction of Sandy and Jerry. Then he raised his eyes again and said, “They were playing catch with a can of tomatoes.” Jerry gasped in indignation, and Sandy quickly gave him a warning nudge. “That’s what they were doing, sir—throwing it back and forth like a couple of schoolkids at a picnic. Then this black-haired fellow here, he let go a good one and it went right through the grandstander’s hands and hit the can of fat on the stove and knocked it over on the fire. And then, sir,” the mate concluded, a note of smugness in his voice, “then, sir, the fat was really in the fire.” With a look of gloating, the captain swung his eyes on Sandy and Jerry—and that was when Sandy opened his mouth and said, “He’s a liar.” Almost the moment that the words dropped from his lips, Sandy Steele wished he could have bitten his tongue in two. But he had finally had to give in to the
- 31. 112 resentment that had been smoldering inside him almost from the moment he had walked aboard the James Kennedy. But, to say that, after all his good advice to Jerry! He glanced over at his friend, half expecting him to be disgusted with him. He was grinning! Then Sandy had to laugh, too—if not from the delight so plain on Jerry’s saucy face, then from the look of injury on the face of the mate. Mr. Briggs actually acted as though he had been unfairly accused! So, Sandy laughed—and when he did, Captain West arose from his chair with a roar of rage. “Get out of here! You smooth-faced, insubordinate little firebugs! Get back to your quarters and stand by to face a court of inquiry on charges of arson and insubordination! That’ll teach you to laugh at me and call my mate a liar! Eh? How about that, eh? How will your friend, Old Man Kennedy, like that, eh, when he hears that his white-faced schoolboys are headed for some Buffalo jail? And you, Mr. Briggs, I’m ordering you to keep these two under lock and key until we get to Buffalo.” Then, puffing up his chest like a giant bullfrog, Captain West issued a final roar: “GET OUT!” Their heads held high, Sandy and Jerry marched back to their quarters. And the door had hardly swung shut behind them, before the skipper whirled and pounced upon his mate with the low snarl of an enraged puma. With a cry and a whimper, the fawning mate who had opened his mouth
- 32. 113 for words of toadying praise, cringed back against the bulkhead. “No, Skipper, don’t,” he whined, but Captain West ignored his pleas and seized him by the shirt collar and began to shake him. “You lying, sniveling drunk!” the skipper growled. “Do you think you fooled me for a moment? I saw you smash that rum bottle in front of that Steele boy’s face tonight. I smelled your breath when you came reeling down the passageway, shrieking like the lily-livered ninny you are.” He shook Mr. Briggs again, fiercely. “Do you think I believed that cock-and-bull story of yours? Do you? Answer me!” Terrified, the mate babbled, “N-no, sir.” “But you still took me for a fool, is that it?” the skipper snarled, almost beside himself. Then, seeing Mr. Briggs burst into a fit of uncontrollable blubbering, he uttered a growl of disgust and flung him back on the bunk like a sack of wheat. He returned to his desk and sat down again. “Briggs,” he said heavily, “if it wasn’t for the fact that I can make use of you, I’d have skinned you alive long ago. I pretended to believe you tonight only because I saw a chance to put those nosy brats of Kennedy’s in their place. I want them under lock and key until that deal is signed in Buffalo. And that’s the day after tomorrow.” The skipper drew another deep breath. “They belong to you, Briggs,” he said. “And you’ll answer for them with your hide.” His voice took on an ugly, menacing tone that raised bumps of fear all along the mate’s spine.
- 33. 114 “If something goes wrong, Briggs, if I see you so much as look at another bottle, I’ll flay that hide of yours from one end of the Lakes to the other. I’ve got too much at stake to fool around! Paul Chadwick wants those Kennedy boats and I want him to get them. If it’s the last thing I do, I’m going to be chief captain of the combined Chadwick and Kennedy lines—and no high school kids are going to get in my way by telling Old Man Kennedy about those high-grade ore discoveries. So, remember that, Briggs—and now get out of here and let me get some sleep.” Still trembling, the shaken mate crept from Captain West’s quarters and closed the door softly behind him. Then he slipped down the passageway toward the tiny cabin occupied by Sandy Steele and Jerry James. The moment Mr. Briggs vanished from sight, the door of the cabin adjoining the skipper’s came stealthily open. Then, slowly, the figure of a little bald-headed man emerged. He shut the door carefully behind him, and then glanced swiftly up and down the corridor. On tiptoe, he slipped over to Captain West’s door. He bent his head to listen. Then he backed off carefully and raised both clenched fists to shake them in a gesture of anger and defiance, before he whirled silently and made his way out of sight. The little bald-headed man was Cookie. He had heard every word spoken in the captain’s cabin since Sandy and Jerry had made their appearance there. Every inch of his little frame burned with determination to come to the rescue of his young friends and help thwart the schemes of the crafty Captain West.
- 34. 115 116 In their own cabin, meanwhile, the two friends had just climbed wearily into their bunks. Suddenly they shot erect as they heard a rattling and clanking outside their door. But they knew in the next instant what the noise meant. It was Mr. Briggs “dogging down” the heavy outside handle. “Well,” Jerry said, “now we’re prisoners.” “Yes,” Sandy said, “but I have a funny feeling that things are going to start to get better.” “Why?” “Because,” Sandy said grimly, “they couldn’t possibly get any worse.”
- 35. 117 CHAPTER TEN The Unsalted Seas Unfortunately, Sandy Steele was wrong. Things could get worse, and they did. They worsened, not only for the two youths from Valley View, California, but for everyone aboard the James Kennedy—to say nothing of all those other thousands of human souls who sailed the lower Lakes on that memorable summer morning. For it was on that morning that a freak summer storm that had been rushing down from the north, roared like a scourge across Lake Huron before bursting in all its fury upon the shallow waters of Lake Erie. It was a storm that blew with shattering force across a body of water notorious for rough weather. There are no storms so sudden and so strong as those that fall upon the Great Lakes, and Sandy Steele and Jerry James were about to witness one of the worst within the memory of the grizzled sailors of “the unsalted seas.” There are the treacherous gales, and sometimes hurricanes, of late fall or early winter—those wailing
- 36. 118 winds that sheathe a ship in fresh-water ice, before driving it to its destruction. In the days of sailing ships, there have been single storms upon the Lakes in which as many as a hundred ships—with thousands of sailors and passengers—have perished within twenty-four hours. Steam-driven freighters, and motorships, too, have sunk to the bottom of these cold waters—and more than a few of the ocean liners that have managed to make their way to the Lakes via the St. Lawrence River have gone to a fresh-water grave. The very first ship to sail the Lakes was the bark, Griffon, of the famous French explorer, LaSalle. It set sail from Buffalo on August 7, 1679, reached the shores of Lake Michigan, and then disappeared completely on its return voyage. From Superior to Ontario, the floors of the Lakes are littered with all manner of ships that have gone down in these storms—with their cargoes, their jewels, their gold, their stacks of currency still undamaged in safes. And it is above the surface of Lake Erie, the body of water toward which the James Kennedy was placidly steaming, that the Great Lakes storms blow the worst and the wildest. For Lake Erie is the shallowest of all the lakes. Its average depth is only 70 feet, compared to that of 250 for the rest of them. At its deepest, it is only 210 feet—compared to 1,180 feet on Lake Superior. Erie is a shallow saucer, a basin, and when the winds go whistling across its surface they create something of the effect that a boy might make by blowing onto a shallow saucer of water—but on a much, much greater scale.
- 37. 119 The winds whip up mountainous waves that can break a freighter in two. There have been storms on Lake Erie as freakish and furious as that recorded by the veteran mariner who had moored his vessel on the Canadian shore opposite Buffalo. To his amazement, the wind blew so savagely that it drove the water out and away from his ship’s hull and left him sitting there, high and dry! Even today, in our modern age, there have been freighters that have ventured into Erie storms, from whom nothing has been heard except a last, despairing message: “We are breaking up.” So it was on Lake Erie that this unusual summer storm struck with such violence, only a few hours after the James Kennedy had left the Detroit River and swung its prow east by north for Buffalo. Oddly enough, Captain West was elated when the storm broke. He would not have been quite so overjoyed had he known how terrible it would become. But his first reaction to the gale was simply that this would probably keep the James Kennedy, and the two youths, out on the Lakes until well after Mr. Paul Chadwick had finished his deal with Mr. Kennedy. In fact, Captain West had decided against going ashore in Detroit for much the same reasons. He had suddenly realized that it might be risky to place Sandy Steele and Jerry James within reach of a big city—with its telephones and telegraphs, and, worse, its buses and railroads. They might, in some way, get off the ship. Then they would be free to warn Mr. Kennedy.
- 38. 120 So Captain West had left orders to make downriver past Detroit and out into Lake Erie. He awoke to the shudder and roll of his ship. In his ears, he could hear the whine of a rising wind. When he gazed out of his porthole, his eyes fell on a slate-gray sea. “A storm!” he cried, grinning with wicked delight. “Oh, ho, Captain West’s luck is running good. This’ll close that deal for good and all!” Pleased as could be, the skipper sprang from his bunk and began putting on his foul-weather clothing. He strode briskly from his cabin. About to make topside, he paused at the mate’s door. He swung it open and leaned in. “Briggs, I think you’d better unlock those boys.” The mate gawked as though he couldn’t believe his ears, but Captain West held up a thick, hairy paw when he opened his mouth to protest. “Do as I say! They’re not going anywhere, especially in this storm. It’s one thing to keep them locked up like that under the pretext of facing charges, Briggs. But it’s another to have them trapped below decks during a storm.” The mate nodded obediently, and Captain West wheeled and headed for the ladder. Moving along the passageway, he was surprised to find that he had to stretch out flat against the bulkhead to keep from falling. The James Kennedy was bucking that much!
- 39. 121 122 Clambering up the ladder, he needed all his strength to keep from being thrown below. When he got on deck, the wind seemed to whistle through his ears, and he pursed his lips in a whistle of his own when he observed the huge, rising seas and the dirty clouds scudding low and threatening above him. Glancing over the side, Captain West whistled again. There was a good two feet less of freeboard already, and the James Kennedy seemed to be plunging deeper into the steely, rain-dimpled waves. Captain West pulled his cap lower on his forehead and thrust one powerful shoulder ahead of him as he bucked into the screaming wind. The rain came slanting at him in sheets and raked his face. He ducked his chin deeper into his shoulder, not quite so jubilant a skipper as he had been upon awakening. For this, indeed, was the start of a real blow! Below decks, Sandy Steele and Jerry James were awake, too. They had been so for perhaps a half hour before Captain West, roused from a deep sleep by the unfamiliar pitching of the vessel. Now they sat on the lower bunk. Both boys had deeply serious expressions on their faces. Sandy was not even aware of the cowlick that hung forward on his forehead, and Jerry James’s brow was a mass of wrinkles. They were listening to the steady clanking and groaning of the James Kennedy’s steel fibers as the laden ore boat rolled in the rising seas. Even below, they could hear the thin wailing of the winds above. “Sounds like a real storm, Sandy.”
- 40. 123 “Yes, and do you realize what this could mean?” “Well, I guess it could mean anything—that is, if it got bad enough.” “Oh, I don’t mean sinking or anything like that. I mean it could keep us from reaching Buffalo in time.” “Oh,” Jerry said, in a small, glum voice, and for a time neither youth spoke. Then they heard a rattling at their door. It opened, and the unfriendly face of Mr. Briggs peeped in. The two youths leaped to their feet. “Stay where you are!” the mate snapped. “You ain’t going anywheres.” He grunted, pushing the door back and securing it against the bulkhead. “Skipper says he wants your door open. Can’t say as I agree with him, but he’s the skipper.” “Can we go out?” Sandy asked. “No.” “How about some food?” Jerry queried, rubbing his stomach. The mate snickered. “You’ll get the same as the others —biscuits and water.” He snickered again. “That’s all the food that’s left after what you two boobs done to the galley.” “What we did!” they chorused, indignantly. “Yes, you!” the mate snarled, backing into the passageway. “And don’t try to come it over me with that
- 41. 124 innocent-angels business.” Sandy and Jerry exchanged glances of amazement, and then, again, they burst out laughing. “Boy, oh, boy,” Jerry breathed, to the annoyance of the mate, “when our Mr. Briggs tells a story, he sticks to it!” The mate’s mouth flew open for an angry reply, but then, it just remained agape and not a sound issued forth. The mate seemed to be rising in the air, towering over the two youths in the cabin. He lost his balance and fell. His mouth still yawning and his hands frantically clawing for a hold on the smooth steel deck, he began to slide toward them. Then the boys were hurled backward against the bulkhead. They struck it with a crash and slithered to the floor, all but stunned. For one long dreadful moment, it seemed to all three of them that the James Kennedy would never return from that sickening roll to starboard. There was that bottomless instant when it appeared that the heavily burdened vessel would never stop heeling over until it had turned turtle and plunged to the bottom. Then, it stopped. It seemed to hang in the air. Sandy and Jerry drew their breath in sharply. They had the terrible sensation that there was nothing beneath the James Kennedy to support it, and that once this
- 42. 125 long, hanging pause had ended—it would drop, drop, drop. Slowly, they let their breath out. The vessel had begun to right itself. With the same slow, deliberate, rolling motion, it heeled over to port, and now it was Sandy and Jerry who rose in the air above the mate and who felt themselves sliding toward him. Again, it seemed that the James Kennedy would overturn, and the hanging sensation was repeated. But when the vessel had righted itself this time, it seemed merely to shiver—before plowing straight ahead. Scrambling erect, the two youths stared at Mr. Briggs. The mate’s face had been drained of color and his little eyes glistened with fear. “That,” he said, in a voice hoarse with awe and disbelief, “was a wave!” Up above, in the pilothouse, Captain West had watched that monster swell come and go, and now even he was a trifle shaken as he mopped his brow in relief. He wondered what would have happened if that wall of water had struck them fore and aft, rather than abeam. He gazed through his windows and wagged his head gravely. The winds still rose in violence. They whipped at the James Kennedy from every quarter, seeming to change direction every other moment like a cyclone gone mad. The seas were a battering confusion. The waves ran this way, the wind another. Between them, they tore at the ship’s superstructure and thundered against her sides. Sometimes two great waves would
- 43. 126 dash at each other from opposite directions, colliding with a great roar and a shattering shower of spray. Captain West saw with alarm that the waves were increasing in height. They were already well past ten feet. They would go on to twenty, of that he was disturbingly certain—and after that? After that, Captain West knew, waves and running seas of that height would batter the long, narrow, shallow James Kennedy until she broke in two. He no longer placed such great importance on staying out of port to make sure of Mr. Chadwick’s deal. He would have given anything, just then, to be safe and snug behind the breakwater at Buffalo. Peering through his rain-splashed windows, the skipper sought a glimpse of some other vessel. But his visibility had been greatly reduced by the sheets of rain and the darkening skies. The unearthly light that had greeted him when he came on deck had been slowly subsiding. Now, as the clock raced on toward noon and the storm raged on in unabated fury, he could see only the clashing seas around him and hear that high-pitched wailing of the wind. He shook himself. “This is bad, very bad,” he said to Sam, who had taken over as wheelsman. “Aye, aye, sir,” Sam said. “I’ve been through some bad ones on the Lakes—but I’ve not seen any worse than this one. And it’s just starting, if I read the signals right.”
- 44. 127 128 The captain bobbed his head in unhappy assent. The James Kennedy staggered and seemed to shake herself as she drove forward into a wall of lake water, and he embraced a stanchion to keep his feet. He waited until the vessel had steadied herself, and then he lurched across the pilothouse to the rear windows to stare with dismay at the spectacle below him. Grayish seas were swamping the decks of the James Kennedy, and the crewmen were frantically at work trying to secure the hatch of one of the holds. Wind and water had torn at a corner of the steel hatch and had peeled it back as though a giant can opener had been at work. Each time the Kennedy dug into one of the heavy seas swinging toward it, the crewmen would seize the rails and hang on for dear life while the water swept down on them. Then, while the vessel rose high again and the waters ran off the sides, they would resume the battle against the hatch—battering away at it with sledge hammers in an attempt to seal the hold. One look at this scene was enough for Captain West. He could see at a glance that more men were needed. “Mr. Briggs!” he shouted at his mate through the speaking tube. “Get every available man up on deck to Number Four hatch!” The mate’s voice wailed hollowly in reply: “They’re all up there already, sir—every man that can be spared.” “Nonsense, Briggs! Who else have you got down there?” “Just myself and those two high school brats.”
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